Eigenvalues

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The eigenvalues of a matrix are the roots of its characteristic equation. The 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 algebraic multiplicity k if (t-c)^k is the highest power of (t-c) that divides the characteristic polynomial.

The algebraic multiplicity of an eigenvalue is greater than or equal to its geometric multiplicity.

Characteristic Equation

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

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

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|.

[n*n]: The characteristic polynomial of A is of the form: tⁿ - tr(A)*tⁿ^-1 + ... + -1ⁿ |A|.
- [2*2]: |tI-A| = t² - tr(A)*t + |A|
[A,B: m*n]: If m>n |tI - AB'| = t^m-n * |tI - B'A|
[n*n]: |tI-AB| = |tI-BA|

Eigenvalues

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.

t is an eigenvalue of A:n*n iff for some non-zero x, Ax=tx. 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.
The eigenvalues of A^k are (eig(A))^k
Similar matrices have the same eigenvalues

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.

The geometric multiplicity of an eigenvalue is less than or equal to its algebraic multiplicity.

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.

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.

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.

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.
If A is nilpotent to index k, its minimal polynomial is t^k.

Singular Values

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

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.