Throughout, \(A\) is an \(n\times n\) matrix over a field \(F\), \(\chi_A(t) = \det(tI-A)\) is its characteristic polynomial, and \(\min_A\) is its minimal polynomial.
The forms and the relations they classify
| Setting | Equivalence relation | Normal form |
|---|---|---|
| \(\chi_A\) splits over \(F\) | similarity | Jordan canonical form |
| any field \(F\) | similarity | rational canonical form |
| matrices over a PID; finitely generated modules | equivalence of matrices; isomorphism of modules | Smith normal form |
| real symmetric or complex normal matrices | orthogonal or unitary similarity | spectral theorem |
The Jordan form of \(A\) over \(F\) exists if and only if \(\chi_A\) splits over \(F\). The rational canonical form exists over every field. Both come from the structure theorem for the finitely generated \(F[t]\)-module \(F^n\) with \(t\) acting by \(A\): the Jordan form from its elementary divisors, the rational form from its invariant factors.
Computation from a matrix
Characteristic polynomial. The roots of \(\chi_A(t) = \det(tI - A)\) are the eigenvalues, with algebraic multiplicities. Also \(\tr A = \sum\lambda_i\) and \(\det A = \prod \lambda_i\), and the eigenvalues of a triangular matrix are its diagonal entries.
Minimal polynomial. \(\min_A\) divides \(\chi_A\) and has the same irreducible factors, so it is determined by the exponents of those factors; it is the monic divisor \(p\) of \(\chi_A\) of least degree, among those with every irreducible factor of \(\chi_A\), such that \(p(A) = 0\).
Diagonalizability. \(A\) is diagonalizable over \(F\) if and only if \(\min_A\) is a product of distinct linear factors over \(F\). In that case \(A\) is similar to \(\diag(\lambda_1,\ldots,\lambda_n)\), with each eigenvalue repeated according to its multiplicity in \(\chi_A\).
Jordan block sizes. For each eigenvalue \(\lambda\):
the exponent of \((t-\lambda)\) in \(\min_A\) is the size of the largest Jordan block for \(\lambda\);
the exponent of \((t-\lambda)\) in \(\chi_A\) is the sum of the sizes of the Jordan blocks for \(\lambda\);
\(\dim\ker(A-\lambda I)\) is the number of Jordan blocks for \(\lambda\).
These three numbers determine the Jordan blocks for \(\lambda\) when its algebraic multiplicity is at most \(6\); for multiplicity \(7\), the block sizes \(3+3+1\) and \(3+2+2\) share all three. In general, \(\dim\ker(A-\lambda I)^k-\dim\ker(A-\lambda I)^{k-1}\) is the number of Jordan blocks for \(\lambda\) of size at least \(k\).
Rational canonical form. If \(\chi_A\) does not split, the rational canonical form is the block diagonal matrix of the companion matrices of the invariant factors \(p_1\divides\cdots\divides p_m\) of \(tI-A\), and \(p_m = \min_A\).
Similarity classes from the two polynomials
Let \(\chi_A = \prod_\lambda (t-\lambda)^{a_\lambda}\) and \(\min_A = \prod_\lambda (t-\lambda)^{m_\lambda}\). The similarity classes of matrices with these polynomials correspond to choices, for each eigenvalue \(\lambda\), of a partition of \(a_\lambda\) with largest part \(m_\lambda\).
For \(\chi_A = (t-2)^4\) and \(\min_A = (t-2)^2\), the partitions of \(4\) with largest part \(2\) are \(2+2\) and \(2+1+1\), so there are two similarity classes, \(J_2(2)\oplus J_2(2)\) and \(J_2(2)\oplus J_1(2)\oplus J_1(2)\).
Polynomial identities satisfied by \(A\)
If \(f(A) = 0\) for a polynomial \(f\), then \(\min_A\) divides \(f\).
| Hypothesis | Consequence for \(\min_A\) |
|---|---|
| \(A^2 = A\) | \(\min_A \divides t^2 - t\), so \(A\) is diagonalizable with eigenvalues in \(\ts{0,1}\) |
| \(A^k = I\) | \(\min_A\divides t^k-1\); if \(\operatorname{char} F \notdivides k\) and \(t^k-1\) splits over \(F\), then \(A\) is diagonalizable over \(F\) |
| \(A^k = 0\) | \(\min_A\divides t^k\), so \(A\) is nilpotent and all its eigenvalues are \(0\) |
In each diagonalizable case, the divisor \(\min_A\) of a polynomial with distinct roots in \(F\) is a product of distinct linear factors.
Consistency conditions
\(\min_A\) and \(\chi_A\) have the same irreducible factors.
For each eigenvalue \(\lambda\), the number of Jordan blocks equals the geometric multiplicity \(\dim\ker(A-\lambda I)\), and the sum of their sizes equals the algebraic multiplicity. The geometric multiplicity is at most the algebraic multiplicity, and \(A\) is diagonalizable over \(F\) if and only if \(\chi_A\) splits and the two multiplicities are equal for every eigenvalue.