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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].

An eigenvalue *c* has an *algebraic multiplicity* (or just *
multiplicity*) of *k* iff
(*t-c*)* ^{k}* is the highest power of (

- The algebraic multiplicity of an eigenvalue is greater than or equal to its geometric multiplicity.
- If, for each of the eigenvalues, the algebraic multiplicity equals the geometric multiplicity, then the matrix is diagonalizable, otherwise it is defective.

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

- [n*n] A matrix
**A**satisfies its own characteristic equation (Cayley-Hamilton theorem)

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

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

- [n*n]: The characteristic polynomial of
**A**is of the form:*t*- tr(^{n}**A**)**t*^{n}^{-1}+ ... + -1|^{n}**A**|.- [2*2]: |
*t***I**-**A**| =*t*^{2}- tr(**A**)**t*+ |**A**|

- [2*2]: |
- [A,B: m*n]: If
*m*>*n*|*t***I**-**AB'**| =*t** |^{m-n}*t***I**-**B'A**| - [n*n]: |
*t***I-AB**| = |*t***I-BA**|

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: |*t***I**-**A**| = 0.

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

*t*is an eigenvalue of**A**:*n*n*iff for some non-zero x,**Ax**=t**x**.**x**is then called an eigenvector corresponding to*t*.- [Complex, n*n]: The matrix
**A**has exactly*n*eigenvalues (not necessarily distinct) - [Complex]: tr(
**A**) = sum(**eig(A)**) - [Complex]: det(
**A**) = prod(**eig(A)**) - [A:m*m, C:n*n]:
**eig([A B; 0 C])**= [**eig(A)**;**eig(C)**] - det(
**A**)=0 iff 0 is an eigenvalue of**A** - The eigenvalues of a triangular or diagonal matrix are its diagonal elements.
- [Hermitian]: The eigenvalues of
**A**are all real. - [Unitary]:
The eigenvalues of
**A**have unit modulus. - [Nilpotent]: The eigenvalues of
**A**are all zero. - [Idempotent]: The eigenvalues of
**A**are all either 0 or 1. - [Normal]: The singular values of
**A**equal the absolute values of the eigenvalues. - [Hermitian +ve semidefinite]: The singular values of
**A**equal the eigenvalues. - The eigenvalues of
**A**are (^{k}**eig(A)**)^{k} - Similar matrices have the same eigenvalues
- [
*Real*, Symmetric, 2#2]**A**=[*a b*;*b d*]=**RDR**^{T}where**R**=[cos(*t*) -sin(*t*); sin(*t*) cos(*t*)] and*t*=½tan^{-1}(2*b*/(*a-d*)) and**D**is diagonal with**D**=**DIAG**([ 1 cos(2*t*) sin(2*t*); 1 -cos(2*t*) -sin(2*t*)] *[ (*a*+*d*)/2; (*a*-*d*)/2;*b*]). The columns of**R**are the eigenvectors and the corresponding elements of**D**the eigenvalues.

- The eigenvectors associated
with an eigenvalue,
*c*, form a subspace whose dimension equals the geometric multiplicity of*c*and never exceeds the algebraic multiplicity of c. This subspace is called the*eigenspace*corresponding to*c*. - A matrix
**A**_{n#n}has an orthonormal set of*n*eigenvectors iff it is normal. In this case we may write**A**=**UDU**^{H}for a diagonal matrix,**D**, of eigenvalues and a unitary matrix,**U**, whose columns are the corresponding eigenvectors. - If
**Ax**=*c***x**. and**A**^{H}**y**=*d***y**where*c*!=*d*, then^{*}**y**^{H}**x**= 0, i.e.**x**and**y**are orthogonal.- [A:Hermitian]: Eigenvectors corresponding to distinct eigenvalues of
**A**are orthogonal.

- [A:Hermitian]: Eigenvectors corresponding to distinct eigenvalues of
- If
**B**and**A**are similar with**B**=**X**^{-1}**AX**for some non-singular**X**, then**y**is an eignvector of**B**iff**Xy**is an eigenvector of**A**corresponding to the same eigenvalue.

- The geometric multiplicity of an eigenvalue is less than or equal to its algebraic multiplicity.
- If, for each of the eigenvalues, the algebraic multiplicity equals the geometric multiplicity, then the matrix is diagonalizable, otherwise it is defective.

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.

- The eigenvalues of
**A**_{[n#n]}lie in the union of the*n*complex-plane closed discs whose centres are**diag**(**A)**and whose radii are**sum**(**ABS**(**A**))**-diag**(**ABS**(**A**)). These discs are the Gersgorin discs (there should be a circumflex over the s of Gersgorin)- Each eigenvalue,
*c*, satisfies min(2**diag**(**ABS**(**A**))**-****sum**(**ABS**(**A**))) <=*|c|*<= max(**sum**(**ABS**(**A**))) - If the
*n*discs can be partitioned into disjoint subsets of the complex plane then each subset contains the same number of (not necessarily distinct) eigenvalues as discs.- [A: real] If the discs are all distinct then the eigenvalues are all real.

- If an eigenvalue has algebraic multiplicity
*m*then it must lie in at least*m*discs.

- Each eigenvalue,

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.

- Similar matrices have the same minimum polynomial.
- The roots of the minimum and characteristic
polynomials are identical (though their multiplicities may differ) and are the
eigenvalues of
**A**. - The minimum polynomial of
**A**_{[n#n]}is a factor of its characteristic polynomial and its order is <=*n*. - The minimum polynomial of a matrix is equal to its characteristic polynomial iff each eigenvalue has a geometric multiplicity of 1.
- If
**A**is nilpotent to index*k*, its minimal polynomial is*t*.^{k}

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

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

- A
*simple*eigenvalue is always regular.

[*m>=n*] The *singular values* of
**A**_{[m#n]} are the positive square roots of the
eigenvalues of
**A**^{H}**A**.

- If
**A**_{[n#n]}is normal, its singular values are the absolute values of its eigenvalues. **A**_{[n#n]}is non-singular iff all its singular values are > 0.- The condition number of a matrix is its largest singular value divided by its smallest singular value.

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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