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The eigenvalues of a square matrix are the roots of its characteristic equation. They may also be referred to by any of the fourteen other combinations of: [characteristic, eigen, latent, proper, secular] + [number, root, value].

Algebraic Multiplicity

An eigenvalue c has an algebraic multiplicity (or just multiplicity) of k iff (t-c)k is the highest power of (t-c) that divides the characteristic polynomial.

Characteristic Equation

[n*n] The characteristic equation of a matrix A is |tI-A| = 0. It is a polynomial equation in t.

Characteristic Matrix

[n*n]: The characteristic matrix of A is (tI-A) and is a function of the scalar t.

Characteristic Polynomial

[n*n] The characteristic polynomial, p(t), of a matrix A is p(t) = |tI - A|.

Defective Eigenvalue

An eigenvalue is defective if its geometric multiplicity is less than its algebraic multiplicity.


The eigenvalues of A are the roots of its characteristic equation: |tI-A| = 0.

The function eig(A) denotes a column vector containing all the eigenvalues of A with appropriate multiplicities.


An eigenvector (also known as a right eigenvector), x, associated with an eigenvalue c of a square matrix A is any vector that satisfies Ax = cx.

Geometric Multiplicity

The geometric multiplicity of an eigenvalue c of a matrix A is the dimension of the subspace of vectors x for which Ax = cx.

Gersgorin Discs

The eigenvalues of a diagonal matrix equal its diagonal elements. If the off-diagonal elements are small rather than being exactly zero, the eigenvalues will be close to the diagonal elements; Gersgorin Discs make this statement precise.

Minimum Polynomial

The minimum polynomial, f(t) of a square matrix A[n#n] is the unique monic polynomial of least degree for which f(A)=0.

Regular Eigenvalue

An eignevalue is regular if its geometric and algebraic multiplicities are equal, otherwise it is defective.

Simple Eigenvalue

An eigenvalue is simple if its algebraic multiplicity is equal to 1. This implies that its geometric multiplicity is also equal to 1.

Singular Values

[m>=n] The singular values of A[m#n] are the positive square roots of the eigenvalues of AHA.

See also: Singular Value Decomposition

This page is part of The Matrix Reference Manual. Copyright © 1998-2022 Mike Brookes, Imperial College, London, UK. See the file gfl.html for copying instructions. Please send any comments or suggestions to "mike.brookes" at "imperial.ac.uk".
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