Throughout, \(A\) is a complex \(n\times n\) matrix, \(A^*\) is its conjugate transpose, and \(\CC^n\) has the standard inner product.
Classes of matrices
| \(A\) is | Definition | Consequences |
|---|---|---|
| Hermitian | \(A^* = A\) | real eigenvalues; eigenvectors for distinct eigenvalues are orthogonal |
| positive definite | \(A^* = A\) and \(x^*Ax > 0\) for all \(x\neq0\) | positive eigenvalues; a Cholesky factorization; a unique positive definite square root |
| unitary | \(A^*A = I\) | eigenvalues of absolute value \(1\); \(\norm{Ax} = \norm x\) for all \(x\) |
| normal | \(A^*A = AA^*\) | unitarily diagonalizable |
Hermitian, skew-Hermitian, and unitary matrices are normal, and by the spectral theorem \(A\) is unitarily diagonalizable if and only if \(A\) is normal.
Factorizations
Spectral decomposition. \(A = U\Lambda U^*\) with \(U\) unitary and \(\Lambda\) diagonal, for normal \(A\).
Singular value decomposition. Every \(m\times n\) complex matrix has a factorization \(A = U\Sigma V^*\) with \(U\) and \(V\) unitary and \(\Sigma\) an \(m\times n\) diagonal matrix whose diagonal entries are the singular values of \(A\), the nonnegative square roots of the eigenvalues of \(A^*A\). The operator norm is \(\norm A_2 = \sigma_{\max}\), the largest singular value.
Cholesky factorization. \(A = LL^*\) with \(L\) lower triangular with positive diagonal entries, for positive definite \(A\).
QR factorization. \(A = QR\) with \(Q\) unitary and \(R\) upper triangular; for invertible \(A\) it is Gram–Schmidt orthonormalization applied to the columns of \(A\).
Eigenvalue location
Gershgorin. Every eigenvalue of \(A=(a_{ij})\) lies in some disc \(\ts{z \st \abs{z-a_{ii}}\leq\sum_{j\neq i}\abs{a_{ij}}}\). A union of \(k\) discs disjoint from the other \(n-k\) discs contains exactly \(k\) eigenvalues, counted with multiplicity.
Courant–Fischer. For Hermitian \(A\) with eigenvalues \(\lambda_1\geq\cdots\geq\lambda_n\), \(\lambda_k = \max_{\dim S = k}\min_{0\neq x\in S} x^*Ax/x^*x\). Hence the eigenvalues of \(A\) and of \(A + vv^*\) interlace.
Trace and determinant. \(\tr A = \sum_i\lambda_i\) and \(\det A = \prod_i\lambda_i\).
Norms
A matrix norm is submultiplicative if \(\norm{AB}\leq\norm A\norm B\) for all \(A,B\); operator norms are submultiplicative.
The entrywise maximum norm \(\norm A_{\max} = \max_{i,j}\abs{a_{ij}}\) is not submultiplicative: for \(A = \matt{1}{1}{1}{1}\), \(\norm{A^2}_{\max} = 2 > 1 = \norm A_{\max}^2\).
For every submultiplicative norm, the spectral radius satisfies \(\rho(A)\leq\norm A\), and \(\rho(A) = \lim_{k\to\infty} \norm{A^k}^{1/k}\) by Gelfand’s formula.