An eigenvalue of a square matrix \(A\) is a scalar \(\lambda\) for which \(Av = \lambda v\) for some nonzero vector \(v\); the corresponding vector \(v\) is the eigenvector. Eigenvalues encode how \(A\) acts along preferred directions and are precisely the roots of the characteristic polynomial \(p(\lambda) = \det(A - \lambda I)\). For a 2×2 matrix \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\) this polynomial becomes \(\lambda^2 - (a+d)\lambda + (ad-bc) = 0\); the sum \(a+d\) is therefore the trace, while the constant \(ad-bc\) is the determinant.
A matrix is diagonalizable when it can be written \(A = P D P^{-1}\) with \(D\) diagonal; the columns of \(P\) are the eigenvectors of \(A\), and \(D\) contains the corresponding eigenvalues. The criterion is clean: diagonalizability holds exactly when \(A\) has \(n\) linearly independent eigenvectors, which happens when the geometric multiplicity of every eigenvalue — the dimension of its eigenspace, the null space of \(A - \lambda I\) — equals its algebraic multiplicity, namely the multiplicity of \(\lambda\) as a root of the characteristic polynomial. The trace equals the sum of eigenvalues, and the Cayley–Hamilton theorem guarantees that every square matrix satisfies its own characteristic polynomial: \(p(A) = 0\).
Several canonical forms sharpen this picture. The spectral theorem says that every real symmetric matrix can be orthogonally diagonalized, \(A = Q D Q^T\) with \(Q\) orthogonal and \(D\) real diagonal. The Schur decomposition relaxes the symmetry requirement to write any square matrix as \(A = Q T Q^*\) with \(Q\) unitary and \(T\) upper triangular; the diagonal entries of \(T\) are the eigenvalues. The polar decomposition goes further by writing a square invertible matrix as \(A = UP\) with \(U\) unitary and \(P\) positive definite. When a matrix is not diagonalizable, the Jordan canonical form provides the closest analogue: every square matrix is similar to an upper-triangular matrix whose diagonal entries are the eigenvalues and whose superdiagonal contains 1s grouped into Jordan blocks. The minimal polynomial is the monic polynomial of least degree annihilating \(A\); for a diagonalizable matrix it is the product \(\prod_i (x - \lambda_i)\) of distinct factors over its eigenvalues. The companion matrix of a monic polynomial \(p(x) = x^n + c_{n-1} x^{n-1} + \cdots + c_0\) is an \(n \times n\) matrix whose characteristic polynomial is \(p\), giving a concrete matrix realization of any monic polynomial.