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    Hoffman and Kunze, Linear Algebra

    SRC-TEXT-HK71

    Area
    Algebra
    Status
    draft
    Source
    Contents

    538 problems.

    1.2

    1. Complex numbers \(x + y\sqrt{2}\) form a subfield of \(\CC\) — Hoffman and Kunze 1.2.1
    2. Equivalence of two systems of two equations over \(\CC\) — Hoffman and Kunze 1.2.2
    3. Testing equivalence of two systems in three unknowns — Hoffman and Kunze 1.2.3
    4. Testing equivalence of two systems in four unknowns over \(\CC\) — Hoffman and Kunze 1.2.4
    5. The two-element field — Hoffman and Kunze 1.2.5
    6. Homogeneous two-unknown systems with the same solutions are equivalent — Hoffman and Kunze 1.2.6
    7. Every subfield of \(\CC\) contains \(\QQ\) — Hoffman and Kunze 1.2.7
    8. Every field of characteristic zero contains a copy of \(\QQ\) — Hoffman and Kunze 1.2.8

    1.3

    1. Solutions to a homogeneous system over \(\CC\) — Hoffman and Kunze 1.3.1
    2. Solving \(AX = 0\) by row reduction — Hoffman and Kunze 1.3.2
    3. Solutions of \(AX = cX\) for eigenvalue equations — Hoffman and Kunze 1.3.3
    4. Row-reducing a matrix over \(\CC\) — Hoffman and Kunze 1.3.4
    5. Non-row-equivalent matrices — Hoffman and Kunze 1.3.5
    6. Row-reduced \(2 \times 2\) complex matrices with trace zero — Hoffman and Kunze 1.3.6
    7. Row interchange from other elementary operations — Hoffman and Kunze 1.3.7
    8. Solutions of \(AX = 0\) for \(2 \times 2\) matrices — Hoffman and Kunze 1.3.8

    1.4

    1. Solving a homogeneous system by row reduction — Hoffman and Kunze 1.4.1
    2. Row-reducing a \(3 \times 2\) matrix over \(\CC\) — Hoffman and Kunze 1.4.2
    3. Classification of \(2 \times 2\) row-reduced echelon matrices — Hoffman and Kunze 1.4.3
    4. Consistency and solutions of a three-equation system — Hoffman and Kunze 1.4.4
    5. Inconsistent system of two equations in two unknowns — Hoffman and Kunze 1.4.5
    6. Inconsistent system of three equations in four unknowns — Hoffman and Kunze 1.4.6
    7. Solving a system of four equations in five unknowns — Hoffman and Kunze 1.4.7
    8. Column space characterization for a \(3 \times 3\) matrix — Hoffman and Kunze 1.4.8
    9. Column space characterization for a \(4 \times 4\) matrix — Hoffman and Kunze 1.4.9
    10. Row-reduced echelon form is unique given the solution set — Hoffman and Kunze 1.4.10

    1.5

    1. Computing two triple products of rectangular matrices — Hoffman and Kunze 1.5.1
    2. Verifying \(A(AB) = A^2B\) directly — Hoffman and Kunze 1.5.2
    3. Non-zero square-zero \(2 \times 2\) matrices — Hoffman and Kunze 1.5.3
    4. Factoring a matrix inverse through elementary matrices — Hoffman and Kunze 1.5.4
    5. Solving \(CA = B\) for a left factor — Hoffman and Kunze 1.5.5
    6. Columns of a product are combinations of the left factor’s columns — Hoffman and Kunze 1.5.6
    7. One-sided inverses of square matrices are two-sided — Hoffman and Kunze 1.5.7
    8. Trace-zero matrices as commutators — Hoffman and Kunze 1.5.8

    1.6

    1. Row-reduction and invertible matrix factorization \(R = PA\) — Hoffman and Kunze 1.6.1
    2. Row-reduction and factorization over \(\CC\) — Hoffman and Kunze 1.6.2
    3. Invertibility via row operations and finding inverses — Hoffman and Kunze 1.6.3
    4. Eigenvectors of a triangular matrix — Hoffman and Kunze 1.6.4
    5. Inverse of a \(4 \times 4\) upper-triangular matrix — Hoffman and Kunze 1.6.5
    6. Product of non-square matrices is not invertible — Hoffman and Kunze 1.6.6
    7. Invertibility and the kernel of a matrix — Hoffman and Kunze 1.6.7
    8. Determinant criterion for \(2 \times 2\) invertibility via row operations — Hoffman and Kunze 1.6.8
    9. Invertibility of upper-triangular matrices — Hoffman and Kunze 1.6.9
    10. \(AB\) not invertible when \(n < m\) — Hoffman and Kunze 1.6.10
    11. Simultaneous row and column reduction to diagonal form — Hoffman and Kunze 1.6.11
    12. The Hilbert matrix is invertible with integer inverse — Hoffman and Kunze 1.6.12

    2.1

    1. \(F^n\) is a vector space over \(F\) — Hoffman and Kunze 2.1.1
    2. A four-vector rearrangement identity in a vector space — Hoffman and Kunze 2.1.2
    3. Span of three vectors in \(\CC^3\) — Hoffman and Kunze 2.1.3
    4. Scalar multiplication that fixes the second coordinate — Hoffman and Kunze 2.1.4
    5. Vector-space axioms under subtraction and negated scalar multiplication — Hoffman and Kunze 2.1.5
    6. Conjugate-symmetric functions form a real vector space — Hoffman and Kunze 2.1.6
    7. Addition and scalar multiplication with outputs on a line — Hoffman and Kunze 2.1.7

    2.2

    1. Subspace tests for subsets of \(\RR^n\) — Hoffman and Kunze 2.2.1
    2. Subspaces of the space of all real functions — Hoffman and Kunze 2.2.2
    3. Membership in the span of three vectors in \(\RR^4\) — Hoffman and Kunze 2.2.3
    4. Spanning set for the solution space of three equations in five unknowns — Hoffman and Kunze 2.2.4
    5. Subspace tests for matrix subsets — Hoffman and Kunze 2.2.5
    6. The subspaces of \(\RR^1\), \(\RR^2\), and \(\RR^3\) — Hoffman and Kunze 2.2.6
    7. A union of two subspaces that is a subspace forces containment — Hoffman and Kunze 2.2.7
    8. Even and odd function subspaces sum to \(V\) with zero intersection — Hoffman and Kunze 2.2.8
    9. Unique decomposition relative to complementary subspaces — Hoffman and Kunze 2.2.9

    2.3

    1. Linear dependence means one vector is a scalar multiple of the other — Hoffman and Kunze 2.3.1
    2. Four vectors in \(\RR^4\): independent or not? — Hoffman and Kunze 2.3.2
    3. Basis for the span of \((1,1,2,4)\), \((2,-1,-5,2)\), \((1,-1,-4,0)\), \((2,1,1,6)\) — Hoffman and Kunze 2.3.3
    4. Three vectors form a basis of \(\RR^3\); express the standard basis in it — Hoffman and Kunze 2.3.4
    5. Pairwise independent but dependent triples in \(\RR^3\) — Hoffman and Kunze 2.3.5
    6. The \(2 \times 2\) matrices have dimension four — Hoffman and Kunze 2.3.6
    7. Dimensions of two matrix subspaces, their sum, and their intersection — Hoffman and Kunze 2.3.7
    8. A basis of idempotent matrices for the \(2 \times 2\) space — Hoffman and Kunze 2.3.8
    9. Pairwise sums of three independent vectors are independent — Hoffman and Kunze 2.3.9
    10. A finitely spanned vector space is finite-dimensional — Hoffman and Kunze 2.3.10
    11. Trace-zero complex matrices as a real vector space, and a conjugation subspace — Hoffman and Kunze 2.3.11
    12. The \(m \times n\) matrices have dimension \(mn\) — Hoffman and Kunze 2.3.12
    13. Pairwise sums of independent vectors over the two-element field — Hoffman and Kunze 2.3.13
    14. \(\RR\) is infinite-dimensional over \(\QQ\) — Hoffman and Kunze 2.3.14

    2.4

    1. Coordinates of the standard basis vectors in a new basis of \(\RR^4\) — Hoffman and Kunze 2.4.1
    2. Coordinate matrix of a vector in a basis of \(\CC^3\) — Hoffman and Kunze 2.4.2
    3. Coordinates of \((a, b, c)\) in a given basis of \(\RR^3\) — Hoffman and Kunze 2.4.3
    4. Two bases of a subspace of \(\CC^3\) and coordinates across them — Hoffman and Kunze 2.4.4
    5. Coordinates in an orthonormal pair of directions in \(\RR^2\) — Hoffman and Kunze 2.4.5
    6. Change of basis between \(\{1, e^{ix}, e^{-ix}\}\) and \(\{1, \cos x, \sin x\}\) — Hoffman and Kunze 2.4.6
    7. Coordinates in the shifted-power basis \(\{1, x+t, (x+t)^2\}\) — Hoffman and Kunze 2.4.7

    2.6

    1. A wide matrix has a non-zero vector in its null space — Hoffman and Kunze 2.6.1
    2. Membership in a real span versus its complex span — Hoffman and Kunze 2.6.2
    3. Homogeneous equations whose solution space is a given span — Hoffman and Kunze 2.6.3
    4. A basis of \(\CC^3\) and coordinates in it — Hoffman and Kunze 2.6.4
    5. Explicit description of a span of four vectors in \(\RR^5\) — Hoffman and Kunze 2.6.5
    6. Basis, membership test, and coordinates for a row space — Hoffman and Kunze 2.6.6
    7. Solvability of \(AX = Y\) via row ranks — Hoffman and Kunze 2.6.7

    3.1

    1. Linearity tests for maps \(\RR^2\to\RR^2\) — Hoffman and Kunze 3.1.1
    2. Range, rank, null space, and nullity of the zero and identity operators — Hoffman and Kunze 3.1.2
    3. Range and null space of differentiation and integration — Hoffman and Kunze 3.1.3
    4. A linear map prescribed on two vectors of \(\RR^3\) — Hoffman and Kunze 3.1.4
    5. A linear map prescribed on three pairs of vectors — Hoffman and Kunze 3.1.5
    6. Explicit form of the operator prescribed on the standard basis — Hoffman and Kunze 3.1.6
    7. Range, rank, null space, and nullity of an operator on \(F^3\) — Hoffman and Kunze 3.1.7
    8. A map with prescribed range spanned by two vectors — Hoffman and Kunze 3.1.8
    9. The commutator map \(A \mapsto AB - BA\) is linear — Hoffman and Kunze 3.1.9
    10. Real-linear but not complex-linear maps on \(\CC\) — Hoffman and Kunze 3.1.10
    11. Left multiplication by \(A\) is the zero map only for \(A = 0\) — Hoffman and Kunze 3.1.11
    12. Range equal to null space forces even dimension — Hoffman and Kunze 3.1.12
    13. Trivial intersection of range and null space versus \(T^2\alpha = 0 \Rightarrow T\alpha = 0\) — Hoffman and Kunze 3.1.13

    3.2

    1. Sums, products, and powers of a reflection and a projection — Hoffman and Kunze 3.2.1
    2. Invertibility of an operator specified on the standard basis of \(\CC^3\) — Hoffman and Kunze 3.2.2
    3. An invertible operator on \(\RR^3\) and a rule for its inverse — Hoffman and Kunze 3.2.3
    4. The operator of Exercise 3 satisfies \((T^2 - I)(T - 3I) = 0\) — Hoffman and Kunze 3.2.4
    5. Rank of left multiplication by a rank-one matrix, and its square — Hoffman and Kunze 3.2.5
    6. A composition through a lower-dimensional space is never invertible — Hoffman and Kunze 3.2.6
    7. Operators with \(TU = 0\) but \(UT \neq 0\) — Hoffman and Kunze 3.2.7
    8. \(T^2 = 0\) relates range to null space; a square-zero example on \(\RR^2\) — Hoffman and Kunze 3.2.8
    9. A one-sided inverse on a finite-dimensional space is the inverse — Hoffman and Kunze 3.2.9
    10. Surjective without injective, and injective without surjective, for rectangular \(A\) — Hoffman and Kunze 3.2.10
    11. Equal ranks of \(T\) and \(T^2\) force range and null space to be disjoint — Hoffman and Kunze 3.2.11
    12. Left multiplication by \(B\) between matrix spaces is invertible only for square invertible \(B\) — Hoffman and Kunze 3.2.12

    3.3

    1. \(\CC\) over \(\RR\) is isomorphic to \(\RR^2\) — Hoffman and Kunze 3.3.1
    2. Reading subspaces of \(V\) off an isomorphism with \(\CC^3\) — Hoffman and Kunze 3.3.2
    3. Hermitian \(2 \times 2\) matrices form a real space isomorphic to \(\RR^4\) — Hoffman and Kunze 3.3.3
    4. \(F^{m \times n}\) is isomorphic to \(F^{mn}\) — Hoffman and Kunze 3.3.4
    5. A multiplicative matrix representation of \(\CC\) over \(\RR\) — Hoffman and Kunze 3.3.5
    6. Finite-dimensional spaces are isomorphic exactly when dimensions agree — Hoffman and Kunze 3.3.6
    7. Conjugation by an isomorphism is an isomorphism of operator spaces — Hoffman and Kunze 3.3.7

    3.4

    1. Matrices of a projection under four basis pairs choices — Hoffman and Kunze 3.4.1
    2. Matrix of a map \(\RR^3 \to \RR^2\) under standard and nonstandard bases — Hoffman and Kunze 3.4.2
    3. The column space of the standard matrix of an operator — Hoffman and Kunze 3.4.3
    4. Every operator on a two-dimensional space satisfies its characteristic polynomial — Hoffman and Kunze 3.4.4
    5. Range and null space bases for an operator given by its standard matrix — Hoffman and Kunze 3.4.5
    6. A rotation has no real characteristic values in any basis — Hoffman and Kunze 3.4.6
    7. An invertible operator on \(\RR^3\) in two bases, with a rule for \(T^{-1}\) — Hoffman and Kunze 3.4.7
    8. The real rotation matrix is similar over \(\CC\) to a diagonal matrix — Hoffman and Kunze 3.4.8
    9. Operators sharing a matrix in some bases are conjugate on \(V\) — Hoffman and Kunze 3.4.9
    10. Classification of idempotent operators on \(\RR^2\) — Hoffman and Kunze 3.4.10
    11. Every operator on a column space is left multiplication; identifying \(UTU^{-1}\)’s matrix — Hoffman and Kunze 3.4.11
    12. Operators of nilpotency index \(n\) share one matrix and are all similar — Hoffman and Kunze 3.4.12
    13. Coordinates in the \(E^{p,q}\) basis of \(L(V, W)\) are the matrix entries of \(T\) — Hoffman and Kunze 3.4.13

    3.5

    1. Linear functionals on \(\RR^3\) prescribed on a basis — Hoffman and Kunze 3.5.1
    2. Dual basis of a given basis of \(\CC^3\) — Hoffman and Kunze 3.5.2
    3. \(\operatorname{trace}(AB) = \operatorname{trace}(BA)\) and similarity invariance of trace — Hoffman and Kunze 3.5.3
    4. Integral functionals on quadratics form the dual of a exhibited basis — Hoffman and Kunze 3.5.4
    5. \(AB - BA = I\) is impossible for complex matrices — Hoffman and Kunze 3.5.5
    6. Every linear map \(F^n \to F^m\) is a tuple of linear functionals — Hoffman and Kunze 3.5.6
    7. Functionals annihilating a span of two vectors in \(\RR^4\) — Hoffman and Kunze 3.5.7
    8. Basis for the annihilator of a span of three vectors in \(\RR^5\) — Hoffman and Kunze 3.5.8
    9. Evaluating an annihilator functional on a given matrix — Hoffman and Kunze 3.5.9
    10. Dimension of the subspace annihilated by \(n\) functionals on \(F^n\) — Hoffman and Kunze 3.5.10
    11. Annihilators reverse sums and intersections of subspaces — Hoffman and Kunze 3.5.11
    12. Linear functionals on a subspace extend to the whole space — Hoffman and Kunze 3.5.12
    13. A pointwise product of functionals that is linear forces a zero factor — Hoffman and Kunze 3.5.13
    14. A functional non-vanishing on finitely many non-zero vectors — Hoffman and Kunze 3.5.14
    15. Trace of left multiplication by \(P\) on \(2 \times 2\) matrices — Hoffman and Kunze 3.5.15
    16. Functionals with \(f(AB) = f(BA)\) are scalar multiples of trace — Hoffman and Kunze 3.5.16
    17. Commutators span exactly the trace-zero matrices — Hoffman and Kunze 3.5.17

    3.6

    1. The annihilator of the sum-zero hyperplane and the dual of \(W\) — Hoffman and Kunze 3.6.1
    2. A subspace is the intersection of the null spaces of its annihilator basis — Hoffman and Kunze 3.6.2
    3. A basis of functions with delta-function evaluation pattern — Hoffman and Kunze 3.6.3

    3.7

    1. Computing the pullback \(T^t f\) for three operators on \(F^2\) — Hoffman and Kunze 3.7.1
    2. The transpose of differentiation applied to an integral functional — Hoffman and Kunze 3.7.2
    3. The transpose of the commutator operator applied to trace — Hoffman and Kunze 3.7.3
    4. Characteristic vectors of \(T\) give characteristic functionals of \(T^t\) — Hoffman and Kunze 3.7.4
    5. Vanishing of \(\operatorname{trace}(A^t A)\) forces \(A = 0\) — Hoffman and Kunze 3.7.5
    6. Null space of the transpose of differentiation on polynomials — Hoffman and Kunze 3.7.6
    7. Transposition is an isomorphism \(L(V, V) \to L(V^*, V^*)\) — Hoffman and Kunze 3.7.7
    8. Every functional on the matrix space is \(A \mapsto \operatorname{trace}(B^t A)\) — Hoffman and Kunze 3.7.8

    4.2

    1. Evaluating polynomials at a \(2 \times 2\) matrix — Hoffman and Kunze 4.2.1
    2. Computing \(f(T)\) for an operator on \(\RR^3\) — Hoffman and Kunze 4.2.2
    3. The characteristic-style polynomial annihilates a diagonal matrix — Hoffman and Kunze 4.2.3
    4. Multiplication by a non-zero polynomial preserves independence — Hoffman and Kunze 4.2.4
    5. \(F^\infty\) has no zero divisors — Hoffman and Kunze 4.2.5
    6. Polynomials of distinct degrees are independent — Hoffman and Kunze 4.2.6
    7. Powers of a linear polynomial form a basis of \(F[x]\) — Hoffman and Kunze 4.2.7
    8. Substitution by \(h\) is injective, surjective only for \(\deg h = 1\) — Hoffman and Kunze 4.2.8
    9. Formal integration and differentiation operators on \(F[x]\) — Hoffman and Kunze 4.2.9

    4.3

    1. Lagrange interpolation through four points — Hoffman and Kunze 4.3.1
    2. Interpolation by quadratics at four points has one compatibility condition — Hoffman and Kunze 4.3.2
    3. Lagrange projections of a diagonal matrix — Hoffman and Kunze 4.3.3
    4. Lagrange resolution of the identity for any operator killed by \(p\) — Hoffman and Kunze 4.3.4
    5. Polynomial evaluation commutes with conjugation — Hoffman and Kunze 4.3.5
    6. Multiplicative linear functionals on \(F[x]\) are evaluations — Hoffman and Kunze 4.3.6

    4.4

    1. Ideals and monic generators in \(\QQ[x]\) — Hoffman and Kunze 4.4.1
    2. Greatest common divisors of three polynomial pairs — Hoffman and Kunze 4.4.2
    3. The polynomials annihilating a matrix form an ideal — Hoffman and Kunze 4.4.3
    4. Minimal polynomial of a triangular matrix — Hoffman and Kunze 4.4.4
    5. Intersections of ideals in \(F[x]\) — Hoffman and Kunze 4.4.5
    6. The ideal generated by finitely many polynomials as an intersection — Hoffman and Kunze 4.4.6
    7. Field extension does not change the g.c.d. of two polynomials — Hoffman and Kunze 4.4.7

    4.5

    1. Cancelling a common monic factor from relatively prime polynomials — Hoffman and Kunze 4.5.1
    2. Coprimality over \(\CC\) means no common root — Hoffman and Kunze 4.5.2
    3. No repeated roots iff \(f\) and \(Df\) are relatively prime — Hoffman and Kunze 4.5.3
    4. Generalized Taylor formula for polynomial composition — Hoffman and Kunze 4.5.4
    5. Congruence modulo a polynomial is an equivalence relation — Hoffman and Kunze 4.5.5
    6. Congruence respects addition and multiplication — Hoffman and Kunze 4.5.6
    7. Polynomial substitution respects congruence — Hoffman and Kunze 4.5.7
    8. Irreducible moduli make polynomial congruence rings integral domains — Hoffman and Kunze 4.5.8

    5.2

    1. Three-linearity tests for functions of \(3\times3\) matrices — Hoffman and Kunze 5.2.1
    2. The three explicit formulas for the \(3 \times 3\) determinant agree — Hoffman and Kunze 5.2.2
    3. The classical adjoint of a \(2 \times 2\) matrix over a commutative ring — Hoffman and Kunze 5.2.3
    4. Invertibility of a \(2 \times 2\) matrix via its determinant — Hoffman and Kunze 5.2.4
    5. Square-zero matrices have characteristic polynomial \(c^2\) — Hoffman and Kunze 5.2.5
    6. A product of entries is n-linear exactly in the distinct-row case — Hoffman and Kunze 5.2.6
    7. The \(2 \times 2\) determinant is alternating and 2-linear over a commutative ring — Hoffman and Kunze 5.2.7
    8. Cofactor expansion defines an alternating 3-linear function — Hoffman and Kunze 5.2.8
    9. Basic row operations leave alternating n-linear functions unchanged — Hoffman and Kunze 5.2.9
    10. Cross-product minors detect rank two and solve the homogeneous system — Hoffman and Kunze 5.2.10
    11. Determinants via alternating 2-linear functions, and multiplicativity — Hoffman and Kunze 5.2.11
    12. Multiplicative matrix functions are non-zero exactly on invertibles — Hoffman and Kunze 5.2.12
    13. Multiplicative functions on \(2 \times 2\) matrices detect singularity — Hoffman and Kunze 5.2.13
    14. The Cayley–Hamilton theorem for \(2 \times 2\) matrices over the ring it generates — Hoffman and Kunze 5.2.14

    5.3

    1. A skew-symmetric \(3 \times 3\) pattern has zero determinant — Hoffman and Kunze 5.3.1
    2. Vandermonde determinant of order three — Hoffman and Kunze 5.3.2
    3. The permutation formula for the \(3 \times 3\) determinant — Hoffman and Kunze 5.3.3
    4. Parity, products, and non-commutativity of two degree-four permutations — Hoffman and Kunze 5.3.4
    5. Invertible matrices have non-zero determinant — Hoffman and Kunze 5.3.5
    6. \(\det(I + A) = 1 + \det A\) exactly for trace zero — Hoffman and Kunze 5.3.6
    7. Determinant of a triangular matrix is its diagonal product — Hoffman and Kunze 5.3.7
    8. Characteristic polynomial of a \(3 \times 3\) matrix; trace and determinant as coefficients — Hoffman and Kunze 5.3.8
    9. Coordinate permutations are invertible linear operators — Hoffman and Kunze 5.3.9
    10. Alternating n-linear functions form a one-dimensional space — Hoffman and Kunze 5.3.10
    11. The determinant scales by a constant under a linear operator — Hoffman and Kunze 5.3.11
    12. Row-permutation matrices, and whether row permutations preserve similarity — Hoffman and Kunze 5.3.12
    13. Uniqueness of the sign function on permutations — Hoffman and Kunze 5.3.13

    5.4

    1. Inverses of two \(3 \times 3\) matrices via the classical adjoint — Hoffman and Kunze 5.4.1
    2. Cramer’s rule for two systems over \(\QQ\) — Hoffman and Kunze 5.4.2
    3. Odd-order complex skew-symmetric matrices are singular — Hoffman and Kunze 5.4.3
    4. Orthogonal matrices have determinant \(\pm 1\) — Hoffman and Kunze 5.4.4
    5. Unitary matrices have determinant of absolute value one — Hoffman and Kunze 5.4.5
    6. Multiplicativity of the operator determinant and invertibility criterion — Hoffman and Kunze 5.4.6
    7. Determinant of a block-diagonal matrix over a commutative ring — Hoffman and Kunze 5.4.7
    8. The commutator operator has determinant zero — Hoffman and Kunze 5.4.8
    9. Determinant rank equals row rank — Hoffman and Kunze 5.4.9
    10. The characteristic equation has at most \(n\) roots in \(F\) — Hoffman and Kunze 5.4.10
    11. \(cA + B\) is invertible for all but at most \(n\) scalars — Hoffman and Kunze 5.4.11
    12. Determinants of left and right multiplication on the matrix space — Hoffman and Kunze 5.4.12
    13. Determinant of \(A \mapsto BAB^*\) on all matrices and on Hermitian matrices — Hoffman and Kunze 5.4.13
    14. Block determinant formula for commuting blocks — Hoffman and Kunze 5.4.14

    6.2

    1. Characteristic polynomials over \(\RR\) and \(\CC\) for three \(2 \times 2\) matrices — Hoffman and Kunze 6.2.1
    2. Characteristic polynomials of the identity and zero operators — Hoffman and Kunze 6.2.2
    3. Characteristic values of a triangular matrix are its diagonal entries — Hoffman and Kunze 6.2.3
    4. Diagonalizing a given operator on \(\RR^3\) — Hoffman and Kunze 6.2.4
    5. Diagonalizability over \(\RR\) versus over \(\CC\) — Hoffman and Kunze 6.2.5
    6. Diagonalizability of a weighted nilpotent shift on \(\RR^4\) — Hoffman and Kunze 6.2.6
    7. Distinct characteristic values imply diagonalizability — Hoffman and Kunze 6.2.7
    8. Invertibility of \(I - BA\) from that of \(I - AB\) — Hoffman and Kunze 6.2.8
    9. \(AB\) and \(BA\) share their characteristic values — Hoffman and Kunze 6.2.9
    10. Real symmetric \(2 \times 2\) matrices are diagonalizable over \(\RR\) — Hoffman and Kunze 6.2.10
    11. Classification of square-zero \(2 \times 2\) complex matrices — Hoffman and Kunze 6.2.11
    12. Every complex \(2 \times 2\) matrix takes one of two canonical shapes — Hoffman and Kunze 6.2.12
    13. The integration operator on continuous functions has no eigenvalues — Hoffman and Kunze 6.2.13
    14. Dimension of the centralizer of a diagonal matrix — Hoffman and Kunze 6.2.14
    15. Characteristic values of a matrix versus its left-multiplication operator — Hoffman and Kunze 6.2.15

    6.3

    1. Minimal polynomials of the identity and zero operators — Hoffman and Kunze 6.3.1
    2. Companion matrix with characteristic and minimal polynomial \(x^3 - ax^2 - bx - c\) — Hoffman and Kunze 6.3.2
    3. A \(4 \times 4\) real matrix with minimal polynomial \(x^2(x-1)^2\) — Hoffman and Kunze 6.3.3
    4. Diagonalizability of a \(4\times4\) matrix with minimal polynomial \(x^2(x-1)^2\) — Hoffman and Kunze 6.3.4
    5. A nilpotent operator on an \(n\)-dimensional space satisfies \(T^n = 0\) — Hoffman and Kunze 6.3.5
    6. A \(3 \times 3\) matrix with minimal polynomial \(x^2\) — Hoffman and Kunze 6.3.6
    7. Minimal polynomial of differentiation on degree-\(n\) polynomials — Hoffman and Kunze 6.3.7
    8. Minimal polynomial of a coordinate projection — Hoffman and Kunze 6.3.8
    9. Trace as a weighted sum of characteristic values — Hoffman and Kunze 6.3.9
    10. Minimal polynomial of left multiplication equals that of the matrix — Hoffman and Kunze 6.3.10
    11. Characteristic and minimal polynomials of \(AB\) versus \(BA\) — Hoffman and Kunze 6.3.11

    6.4

    1. Invariant subspaces of an operator over \(\RR\) versus over \(\CC\) — Hoffman and Kunze 6.4.1
    2. The minimal polynomial of a restriction divides the minimal polynomial — Hoffman and Kunze 6.4.2
    3. Restriction to an eigenspace — Hoffman and Kunze 6.4.3
    4. Triangulating a given \(3 \times 3\) real matrix — Hoffman and Kunze 6.4.4
    5. Idempotent matrices are diagonalizable — Hoffman and Kunze 6.4.5
    6. Restrictions of diagonalizable operators to invariant subspaces — Hoffman and Kunze 6.4.6
    7. Diagonalizability over \(\CC\) via polynomials with distinct roots — Hoffman and Kunze 6.4.7
    8. Operators leaving every subspace invariant are scalars — Hoffman and Kunze 6.4.8
    9. Invariant subspaces of the indefinite integration operator — Hoffman and Kunze 6.4.9
    10. Real matrices either triangulate over \(\RR\) or diagonalize over \(\CC\) — Hoffman and Kunze 6.4.10
    11. Triangular and similar to diagonal implies already diagonal? — Hoffman and Kunze 6.4.11
    12. Spectral mapping for polynomials over an algebraically closed field — Hoffman and Kunze 6.4.12
    13. Diagonalizability of left multiplication and the commutator map — Hoffman and Kunze 6.4.13

    6.5

    1. Simultaneous diagonalization of two real matrices — Hoffman and Kunze 6.5.1
    2. Dimension bounds for commuting families of complex matrices — Hoffman and Kunze 6.5.2
    3. Operators commuting with one having \(n\) distinct eigenvalues are polynomials in it — Hoffman and Kunze 6.5.3
    4. Block determinant formula for commuting blocks over \(\CC\) — Hoffman and Kunze 6.5.4
    5. Commutator operators by diagonal matrices are simultaneously diagonalizable — Hoffman and Kunze 6.5.5

    6.6

    1. Every subspace is a direct summand — Hoffman and Kunze 6.6.1
    2. Dimension counting makes a spanning sum a direct sum — Hoffman and Kunze 6.6.2
    3. Projection onto one line along another in \(\RR^2\) — Hoffman and Kunze 6.6.3
    4. Sums of projections onto independent subspaces — Hoffman and Kunze 6.6.4
    5. Polynomials in a projection collapse to \(aI + bE\) — Hoffman and Kunze 6.6.5
    6. Diagonalizable with spectrum \(\{0, 1\}\) implies projection? — Hoffman and Kunze 6.6.6
    7. The complementary projection \(I - E\) — Hoffman and Kunze 6.6.7
    8. Resolution of the identity by pairwise orthogonal projections — Hoffman and Kunze 6.6.8
    9. Inverting \(I + E\) for a projection \(E\) — Hoffman and Kunze 6.6.9
    10. A resolution of the identity by projections has pairwise zero products — Hoffman and Kunze 6.6.10
    11. The dual space decomposes as the direct sum of annihilators — Hoffman and Kunze 6.6.11

    6.7

    1. Invariance of a projection’s range and null space versus commuting — Hoffman and Kunze 6.7.1
    2. A Jordan block’s eigenline admits no invariant complement — Hoffman and Kunze 6.7.2
    3. Range and null space are independent exactly when they span \(V\) — Hoffman and Kunze 6.7.3
    4. Determinant, characteristic polynomial, and minimal polynomial of an invariant direct sum — Hoffman and Kunze 6.7.4
    5. Spectral projections of the Example 6.2.3 operator via Lagrange polynomials — Hoffman and Kunze 6.7.5
    6. Spectral resolution of the Example 6.3.5 matrix — Hoffman and Kunze 6.7.6
    7. Eigenspaces versus columns of the other spectral projections — Hoffman and Kunze 6.7.7
    8. Operators commuting with every projection — Hoffman and Kunze 6.7.8
    9. Even/odd decomposition of continuous functions and invariance under integration — Hoffman and Kunze 6.7.9

    6.8

    1. Primary decomposition of a given operator on \(\RR^3\) — Hoffman and Kunze 6.8.1
    2. Jordan decomposition of a given operator on \(\RR^3\) — Hoffman and Kunze 6.8.2
    3. Differentiation on polynomials of degree at most \(n\) is nilpotent — Hoffman and Kunze 6.8.3
    4. The primary component is the set of generalized eigenvectors, of dimension \(d_i\) — Hoffman and Kunze 6.8.4
    5. The diagonalizable part of \(g(T)\) is \(g(D)\) — Hoffman and Kunze 6.8.5
    6. Rank-one operators are diagonalizable or nilpotent, not both — Hoffman and Kunze 6.8.6
    7. Operators commuting with all diagonalizable operators are scalars — Hoffman and Kunze 6.8.7
    8. Nilpotent \(A\) makes the commutator operator nilpotent — Hoffman and Kunze 6.8.8
    9. Two non-similar nilpotent \(4 \times 4\) matrices with one minimal polynomial — Hoffman and Kunze 6.8.9
    10. Invariant subspaces split along the primary decomposition — Hoffman and Kunze 6.8.10
    11. Failure of the triangular Jordan-decomposition argument — Hoffman and Kunze 6.8.11
    12. A basis adapted to the commuting diagonalizable and nilpotent parts — Hoffman and Kunze 6.8.12
    13. A vector whose \(T\)-annihilator is the full primary power \(p^n\) — Hoffman and Kunze 6.8.13
    14. Some vector has the minimal polynomial as its \(T\)-annihilator — Hoffman and Kunze 6.8.14
    15. Nilpotent operators have characteristic polynomial \(x^n\) — Hoffman and Kunze 6.8.15

    7.1

    1. Non-eigenvectors in \(\FF^2\) are cyclic vectors — Hoffman and Kunze 7.1.1
    2. A diagonal operator with a repeated value has no cyclic vector — Hoffman and Kunze 7.1.2
    3. \(T\)-annihilators of two vectors for a given operator on \(\CC^3\) — Hoffman and Kunze 7.1.3
    4. A cyclic vector for \(T^2\) gives one for \(T\) — Hoffman and Kunze 7.1.4
    5. Vectors outside the kernel of \(N^{n-1}\) are cyclic for nilpotent \(N\) — Hoffman and Kunze 7.1.5
    6. Direct proof that the companion matrix has \(p\) as characteristic polynomial — Hoffman and Kunze 7.1.6
    7. Cyclic vectors and distinct eigenvalues for diagonalizable operators — Hoffman and Kunze 7.1.7
    8. Commutants of operators with cyclic vectors — Hoffman and Kunze 7.1.8

    7.2

    1. A cyclic subspace with no disjoint cyclic subspace — Hoffman and Kunze 7.2.1
    2. Complementary invariant subspaces for the range — Hoffman and Kunze 7.2.2
    3. An eigenspace with no complementary invariant subspace — Hoffman and Kunze 7.2.3
    4. Annihilator ideals inside the eigenspace of a Jordan block — Hoffman and Kunze 7.2.4
    5. Applying a polynomial distributes over direct sums of invariant subspaces — Hoffman and Kunze 7.2.5
    6. Equal annihilators are preserved by applying polynomials — Hoffman and Kunze 7.2.6
    7. Minimal polynomials and rational forms of three real matrices — Hoffman and Kunze 7.2.7
    8. Cyclic decomposition vectors for a given operator on \(\RR^3\) — Hoffman and Kunze 7.2.8
    9. Rationalizing a given real matrix — Hoffman and Kunze 7.2.9
    10. Minimal polynomials of a one-parameter Jordan family in three cases — Hoffman and Kunze 7.2.10
    11. Characteristic plus minimal polynomial determine \(3 \times 3\) similarity, not \(4 \times 4\) — Hoffman and Kunze 7.2.11
    12. Similarity over \(\CC\) implies similarity over the subfield — Hoffman and Kunze 7.2.12
    13. Real spectrum forces real similarity — Hoffman and Kunze 7.2.13
    14. Separating vectors for the polynomial algebra of an operator — Hoffman and Kunze 7.2.14
    15. Minimal polynomial is stable under scalar field extension — Hoffman and Kunze 7.2.15
    16. Matrices squaring to \(-I\) have even order and a standard block form — Hoffman and Kunze 7.2.16
    17. Primary operators with cyclic minimal polynomial admit no invariant complements — Hoffman and Kunze 7.2.17
    18. Invariant subspaces of diagonalizable operators have invariant complements — Hoffman and Kunze 7.2.18
    19. Cyclic vector iff the commutant of \(T\) consists of polynomials in \(T\) — Hoffman and Kunze 7.2.19
    20. All non-zero vectors are cyclic iff the characteristic polynomial is irreducible — Hoffman and Kunze 7.2.20
    21. Irreducible real action forces complex diagonalizability — Hoffman and Kunze 7.2.21

    7.3

    1. Nilpotent \(3 \times 3\) matrices are classified by their minimal polynomials — Hoffman and Kunze 7.3.1
    2. Characteristic plus minimal polynomial determine similarity when \(d_i \leq 3\) — Hoffman and Kunze 7.3.2
    3. Jordan form from given characteristic and minimal polynomials — Hoffman and Kunze 7.3.3
    4. Counting Jordan forms for a \(6 \times 6\) spectrum — Hoffman and Kunze 7.3.4
    5. Jordan form of the differentiation matrix on quartics — Hoffman and Kunze 7.3.5
    6. Jordan form of a given \(6 \times 6\) complex matrix — Hoffman and Kunze 7.3.6
    7. Trace from the characteristic polynomial factorization — Hoffman and Kunze 7.3.7
    8. Similarity classes of cube roots of the identity — Hoffman and Kunze 7.3.8
    9. Similarity classes of \(n\)th roots of the identity — Hoffman and Kunze 7.3.9
    10. A nilpotent of maximal index has no square root — Hoffman and Kunze 7.3.10
    11. Nilpotent \(6 \times 6\) matrices classified by minimal polynomial and nullity — Hoffman and Kunze 7.3.11
    12. Characteristic, minimal polynomial, and eigenspace dimensions determine similarity when \(d_i \leq 6\) — Hoffman and Kunze 7.3.12
    13. Every complex matrix is similar to its transpose — Hoffman and Kunze 7.3.13
    14. Finding the flaw in a skew-symmetric-matrix proof, with counterexample — Hoffman and Kunze 7.3.14
    15. Square roots of unipotent matrices via the binomial series — Hoffman and Kunze 7.3.15
    16. Every non-singular complex matrix has a square root — Hoffman and Kunze 7.3.16

    7.4

    1. Row reduction to upper-triangular form over \(F[x]\) — Hoffman and Kunze 7.4.1
    2. Cyclic vector criterion via subresultant determinants of \(xI - A\) — Hoffman and Kunze 7.4.2
    3. First invariant factor of \(xI-A\) — Hoffman and Kunze 7.4.3
    4. Constructing an operator with prescribed characteristic and minimal polynomials — Hoffman and Kunze 7.4.4
    5. Full invariant-factor and Jordan analysis of an \(8 \times 8\) matrix — Hoffman and Kunze 7.4.5

    7.5

    1. The semi-simple part of any polynomial in a nilpotent operator is scalar — Hoffman and Kunze 7.5.1
    2. Polynomials in a semi-simple operator are semi-simple — Hoffman and Kunze 7.5.2
    3. Semi-simple operators annihilate no non-zero nilpotent polynomial values — Hoffman and Kunze 7.5.3

    8.1

    1. Inner products vanish on zero and separate points — Hoffman and Kunze 8.1.1
    2. Sums and positive multiples of inner products — Hoffman and Kunze 8.1.2
    3. All inner products on \(\RR^1\) and \(\CC^1\) — Hoffman and Kunze 8.1.3
    4. The standard inner product on \(F^n\) — Hoffman and Kunze 8.1.4
    5. Solving inner-product conditions and expanding vectors in \(\RR^2\) — Hoffman and Kunze 8.1.5
    6. Inner products making rotation through \(90^\circ\) orthogonal — Hoffman and Kunze 8.1.6
    7. No non-zero operator on \(\CC^2\) is everywhere imaginary-orthogonal — Hoffman and Kunze 8.1.7
    8. Matrix criteria for \(Y^tAX\) to be an inner product — Hoffman and Kunze 8.1.8
    9. Inner-product norms satisfy the parallelogram law — Hoffman and Kunze 8.1.9
    10. Gram matrix of the Example 2 inner product — Hoffman and Kunze 8.1.10
    11. The Hilbert-matrix inner product on polynomials — Hoffman and Kunze 8.1.11
    12. Prescribing all inner products with a basis at once — Hoffman and Kunze 8.1.12
    13. Conjugations and their fixed real part — Hoffman and Kunze 8.1.13
    14. Real forms with unique real and imaginary parts define conjugations — Hoffman and Kunze 8.1.14
    15. All conjugations on \(\CC^1\) and \(\CC^2\) — Hoffman and Kunze 8.1.15
    16. Real subspaces meeting every complex basis — Hoffman and Kunze 8.1.16
    17. Extending an inner product from the fixed set of a conjugation — Hoffman and Kunze 8.1.17

    8.2

    1. The joint orthogonal complement of two vectors in \(\RR^4\) — Hoffman and Kunze 8.2.1
    2. Gram–Schmidt on three vectors in \(\RR^3\) — Hoffman and Kunze 8.2.2
    3. Orthonormal basis for a span of two vectors in \(\CC^3\) — Hoffman and Kunze 8.2.3
    4. The norm distance is a metric — Hoffman and Kunze 8.2.4
    5. Vectors agreeing against every tester are equal — Hoffman and Kunze 8.2.5
    6. Orthogonal projection onto a line in the standard plane — Hoffman and Kunze 8.2.6
    7. Orthogonal projection in a non-standard inner product — Hoffman and Kunze 8.2.7
    8. An inner product with prescribed value on two vectors — Hoffman and Kunze 8.2.8
    9. Orthogonal polynomials on \([0,1]\) via Gram–Schmidt — Hoffman and Kunze 8.2.9
    10. Orthogonal complement of the diagonal matrices under the trace pairing — Hoffman and Kunze 8.2.10
    11. Expanding inner products over an orthonormal basis — Hoffman and Kunze 8.2.11
    12. Orthogonal projections are self-adjoint — Hoffman and Kunze 8.2.12
    13. Double orthogonal complements recover the span — Hoffman and Kunze 8.2.13
    14. Matrix entries of an operator in an orthonormal basis — Hoffman and Kunze 8.2.14
    15. Gluing inner products across a direct-sum decomposition — Hoffman and Kunze 8.2.15
    16. Norm-nonincreasing projections are the orthogonal one — Hoffman and Kunze 8.2.16
    17. Orthogonal complement of the odd functions on \([-1, 1]\) — Hoffman and Kunze 8.2.17

    8.3

    1. Computing the adjoint operator on \(\CC^2\) — Hoffman and Kunze 8.3.1
    2. The adjoint matrix and whether \(T\) is normal — Hoffman and Kunze 8.3.2
    3. Null space of the adjoint of an operator with power-matrix entries — Hoffman and Kunze 8.3.3
    4. The adjoint’s range is the orthogonal complement of the null space — Hoffman and Kunze 8.3.4
    5. Inverting the adjoint — Hoffman and Kunze 8.3.5
    6. Rank-one operators \((\alpha \mid \beta)\gamma\) and their adjoints — Hoffman and Kunze 8.3.6
    7. Products of self-adjoint operators — Hoffman and Kunze 8.3.7
    8. The reproducing kernel polynomial for evaluation on \([0,1]\) — Hoffman and Kunze 8.3.8
    9. Adjoint of differentiation under the \([0,1]\) integral pairing — Hoffman and Kunze 8.3.9
    10. Adjoint of conjugation \(A \mapsto P^{-1}AP\) under the trace pairing — Hoffman and Kunze 8.3.10
    11. Idempotent operators commuting with their adjoints are self-adjoint — Hoffman and Kunze 8.3.11
    12. Self-adjoint operators characterized by real quadratic values — Hoffman and Kunze 8.3.12

    8.4

    1. Unitary versus orthogonal matrices — Hoffman and Kunze 8.4.1
    2. Left multiplication on the matrix space is unitary iff the factor is — Hoffman and Kunze 8.4.2
    3. Multiplication operators on \(\CC\) as a real inner product space — Hoffman and Kunze 8.4.3
    4. The plane orthogonal group is rotations and reflected rotations — Hoffman and Kunze 8.4.4
    5. Matrix of a rotation about a prescribed axis in \(\RR^3\) — Hoffman and Kunze 8.4.5
    6. Reflections across subspaces are self-adjoint and unitary — Hoffman and Kunze 8.4.6
    7. The Cayley transform of a self-adjoint operator — Hoffman and Kunze 8.4.7
    8. Rotation matrices are unitarily equivalent to their diagonal form — Hoffman and Kunze 8.4.8
    9. Unitarity under a positive reparametrized inner product — Hoffman and Kunze 8.4.9
    10. The rank-one operators \(T_{\alpha,\beta}\) and their algebra — Hoffman and Kunze 8.4.10
    11. The operator norm induced by rank-one operators, and its matrix model — Hoffman and Kunze 8.4.11
    12. Every self-adjoint unitary operator is a reflection across a subspace — Hoffman and Kunze 8.4.12
    13. Conjugation preserves adjoints, traces, and rank-one operators — Hoffman and Kunze 8.4.13

    8.5

    1. Orthogonal diagonalization of three real symmetric matrices — Hoffman and Kunze 8.5.1
    2. Complex symmetric matrices: self-adjoint? normal? — Hoffman and Kunze 8.5.2
    3. Orthogonal diagonalization of a \(3 \times 3\) symmetric matrix — Hoffman and Kunze 8.5.3
    4. Spectral basis for a normal operator on \(\CC^2\) — Hoffman and Kunze 8.5.4
    5. A non-normal matrix whose square is normal — Hoffman and Kunze 8.5.5
    6. Normal operators classified by their spectra — Hoffman and Kunze 8.5.6
    7. Positive and unitary forces the identity — Hoffman and Kunze 8.5.7
    8. Normal operators as commuting real and imaginary self-adjoint parts — Hoffman and Kunze 8.5.8
    9. Real symmetric matrices have real symmetric cube roots — Hoffman and Kunze 8.5.9
    10. Positive matrices have positive square roots — Hoffman and Kunze 8.5.10
    11. Normal and nilpotent implies zero — Hoffman and Kunze 8.5.11
    12. Eigenvectors of normal operators for distinct eigenvalues are orthogonal — Hoffman and Kunze 8.5.12
    13. The adjoint of a normal operator is a polynomial in it — Hoffman and Kunze 8.5.13
    14. Products of commuting normal operators are normal — Hoffman and Kunze 8.5.14

    9.2

    1. Sesquilinear forms on \(\CC^2\) — Hoffman and Kunze 9.2.1
    2. Matrices of a bilinear form in three bases of \(\RR^2\) — Hoffman and Kunze 9.2.2
    3. A Hermitian-matrix inner product on \(\CC^2\) — Hoffman and Kunze 9.2.3
    4. Symmetric sesquilinear forms are zero — Hoffman and Kunze 9.2.4
    5. Diagonalizing a symmetric bilinear form on \(\RR^2\) — Hoffman and Kunze 9.2.5
    6. Non-degeneracy of a form versus its associated operator — Hoffman and Kunze 9.2.6
    7. Left non-degeneracy equals right non-degeneracy — Hoffman and Kunze 9.2.7
    8. Representing functionals by vectors through a non-degenerate form — Hoffman and Kunze 9.2.8
    9. Adjoints relative to a non-degenerate form — Hoffman and Kunze 9.2.9

    9.3

    1. Images of \(\epsilon_1\) under positive operators — Hoffman and Kunze 9.3.1
    2. Positive plane rotations — Hoffman and Kunze 9.3.2
    3. Adjoints and representing vectors under a \(G\)-weighted inner product — Hoffman and Kunze 9.3.3
    4. Sums of positive operators are positive; products need not be — Hoffman and Kunze 9.3.4
    5. Positivity of a \(2 \times 2\) matrix and the induced inner product geometry — Hoffman and Kunze 9.3.5
    6. Positivity tests for four matrices — Hoffman and Kunze 9.3.6
    7. Positive principal minors do not imply a positive matrix — Hoffman and Kunze 9.3.7
    8. An inner-product test on \(\CC^2\) — Hoffman and Kunze 9.3.8
    9. Diagonal entries of positive matrices — Hoffman and Kunze 9.3.9
    10. The order relation on self-adjoint operators — Hoffman and Kunze 9.3.10
    11. Positivity of \(cI + E\) and a square root of \(I + E\) — Hoffman and Kunze 9.3.11
    12. The Hilbert matrix is positive — Hoffman and Kunze 9.3.12
    13. Shifting a self-adjoint matrix into the positive cone — Hoffman and Kunze 9.3.13
    14. Products of positive operators are positive exactly under commutation — Hoffman and Kunze 9.3.14
    15. Characteristic values of products of positive operators — Hoffman and Kunze 9.3.15

    9.5

    1. Non-negative matrices have unique non-negative square roots — Hoffman and Kunze 9.5.1
    2. The exponential of a skew-adjoint matrix is unitary — Hoffman and Kunze 9.5.2
    3. Sums and products of commuting normal operators are normal — Hoffman and Kunze 9.5.3
    4. Ten equivalent characterizations of normality — Hoffman and Kunze 9.5.4
    5. Commuting normal operators generate a self-adjoint algebra of normal operators — Hoffman and Kunze 9.5.5
    6. The Cayley transform between unitary and self-adjoint operators — Hoffman and Kunze 9.5.6
    7. Spectral resolutions of the left, right, and commutator families — Hoffman and Kunze 9.5.7
    8. Left multiplication by \(B\) is unitarily equivalent to right multiplication by \(B^t\) — Hoffman and Kunze 9.5.8
    9. Conjugation operators of the unitary group and their non-realizability — Hoffman and Kunze 9.5.9
    10. Roots of a generated self-adjoint algebra are multiplicative linear functions — Hoffman and Kunze 9.5.10
    11. Roots of the algebra generated by a commuting family extend family roots uniquely — Hoffman and Kunze 9.5.11
    12. Comparing the algebras generated by a commuting family with and without \(I\) — Hoffman and Kunze 9.5.12

    10.1

    1. Bilinear forms on \(\RR^2\) — Hoffman and Kunze 10.1.1
    2. Matrices of a bilinear form on \(\RR^2\) in three bases — Hoffman and Kunze 10.1.2
    3. The trace pairing on \(2 \times 3\) matrices against a fixed matrix — Hoffman and Kunze 10.1.3
    4. Symmetric bilinear forms on \(\RR^3\) — Hoffman and Kunze 10.1.4
    5. Skew-symmetric bilinear forms on \(\RR^3\) — Hoffman and Kunze 10.1.5
    6. A trace-based bilinear form on the full matrix algebra — Hoffman and Kunze 10.1.6
    7. Degeneracy on the full matrix space, non-degeneracy on trace zero — Hoffman and Kunze 10.1.7
    8. The trace form restricted to skew-Hermitian trace-zero matrices — Hoffman and Kunze 10.1.8
    9. The radical of the trace form — Hoffman and Kunze 10.1.9
    10. Rank of a bilinear form via its right radical — Hoffman and Kunze 10.1.10
    11. Non-degenerate restrictions bound the rank below — Hoffman and Kunze 10.1.11
    12. Factoring forms through a non-singular form — Hoffman and Kunze 10.1.12
    13. The factorization of Exercise 12 fails for singular \(g\) — Hoffman and Kunze 10.1.13
    14. Rank-one bilinear forms are exactly products of two functionals — Hoffman and Kunze 10.1.14

    10.2

    1. Recovering the symmetric form from seven quadratic forms — Hoffman and Kunze 10.2.1
    2. Matrices, ranks, and non-degeneracy of seven quadratic forms on \(\mathbb R^2\) — Hoffman and Kunze 10.2.2
    3. Non-degeneracy of a binary quadratic form via its discriminant — Hoffman and Kunze 10.2.3
    4. The space of symmetric forms, quadratic forms, and their pullback operators — Hoffman and Kunze 10.2.4
    5. Completing the square as a change of variables — Hoffman and Kunze 10.2.5
    6. Diagonalizing the pure cross term \(2bx_1x_2\) — Hoffman and Kunze 10.2.6
    7. Diagonalizing a ternary quadratic form by change of variables — Hoffman and Kunze 10.2.7
    8. Every real quadratic form reduces to a signed sum of squares — Hoffman and Kunze 10.2.8
    9. Diagonal matrices for symmetric forms on \(\RR^n\) — Hoffman and Kunze 10.2.9
    10. The determinant as a quadratic form on Hermitian \(2 \times 2\) matrices — Hoffman and Kunze 10.2.10
    11. Adjoints relative to a non-degenerate symmetric bilinear form — Hoffman and Kunze 10.2.11
    12. Matrix formula for an adjoint relative to \(X^tAY\) — Hoffman and Kunze 10.2.12
    13. The basis dual to a basis under a non-degenerate symmetric form — Hoffman and Kunze 10.2.13
    14. Relative-adjoint matrices in an \(f\)-dual basis — Hoffman and Kunze 10.2.14
    15. Coordinates of the \(f\)-dual basis from the inverse Gram matrix — Hoffman and Kunze 10.2.15
    16. Matrix formula for an \(f\)-dual column basis — Hoffman and Kunze 10.2.16
    17. Orthogonal complements for a symmetric bilinear form — Hoffman and Kunze 10.2.17
    18. Self-dual bases for non-degenerate complex symmetric forms — Hoffman and Kunze 10.2.18

    10.3

    1. Skew-symmetric forms form a subspace of the form space — Hoffman and Kunze 10.3.1
    2. All skew-symmetric bilinear forms on \(\RR^3\) — Hoffman and Kunze 10.3.2
    3. A basis for the skew-symmetric forms on \(\RR^n\) — Hoffman and Kunze 10.3.3
    4. Symmetric and skew-symmetric parts intersect only at zero — Hoffman and Kunze 10.3.4
    5. The symmetrizing projection on the space of bilinear forms — Hoffman and Kunze 10.3.5
    6. Non-degenerate restrictions of skew-symmetric forms — Hoffman and Kunze 10.3.6
    7. Skew-symmetry characterized by the left and right maps — Hoffman and Kunze 10.3.7
    8. Diagonalization analogue for skew-symmetric forms — Hoffman and Kunze 10.3.8
    9. Wedge products of two functionals as skew forms — Hoffman and Kunze 10.3.9
    10. Rank-two skew-symmetric forms as wedges of independent functionals — Hoffman and Kunze 10.3.10
    11. Every skew form on \(\RR^3\) is a wedge of two functionals — Hoffman and Kunze 10.3.11
    12. Congruence of skew-symmetric forms is rank alone — Hoffman and Kunze 10.3.12
    13. Congruence by rank for complex symmetric forms, and its real failure — Hoffman and Kunze 10.3.13

    10.4

    1. Closure of the complex orthogonal group under transpose, conjugate, and adjoint — Hoffman and Kunze 10.4.1
    2. Similarity and the complex orthogonal group — Hoffman and Kunze 10.4.2
    3. Orthogonal changes of variables preserve the sum of squares — Hoffman and Kunze 10.4.3
    4. Orthogonality of a matrix via its columns — Hoffman and Kunze 10.4.4
    5. First columns of complex orthogonal matrices — Hoffman and Kunze 10.4.5
    6. Completing an orthogonal matrix from a prescribed row — Hoffman and Kunze 10.4.6
    7. The dot-product form in an orthogonal basis — Hoffman and Kunze 10.4.7
    8. Carrying a unit vector to a coordinate vector by orthogonal matrices — Hoffman and Kunze 10.4.8
    9. Forms invariant under the complex orthogonal group — Hoffman and Kunze 10.4.9
    10. The commutant of a set of matrices is an algebra — Hoffman and Kunze 10.4.10
    11. Form-preserving operators have determinant \(\pm 1\) — Hoffman and Kunze 10.4.11
    12. The matrix condition for preserving \(X^tAY\) — Hoffman and Kunze 10.4.12
    13. Operators preserving a twisted form \(g(\alpha, T\beta)\) — Hoffman and Kunze 10.4.13
    14. The Lorentz group of \(x_1^2 - x_2^2\) over \(\CC\) — Hoffman and Kunze 10.4.14
    15. The area form on \(\CC^2\) and its preservation group — Hoffman and Kunze 10.4.15
    16. Conditions for a block matrix to lie in the complex symplectic group — Hoffman and Kunze 10.4.16
    17. Forms invariant under the real orthogonal group — Hoffman and Kunze 10.4.17
    18. Forms invariant under the complex orthogonal group — Hoffman and Kunze 10.4.18

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