Eigenvalues

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)*t^n-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. 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(B)]
• 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.

• [n*n]: The eigenvalues of A lie in the union of the n complex-plane 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)
  • [n*n]: All eigenvalues of a matrix have absolute values in the range min(sum(ABS(A))) to max(sum(ABS(A)))
• [n*n]: 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.

Minimum Polynomial

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

• [n*n]: Similar matrices have the same minimum polynomial.
• [n*n]: The roots of the minimum and characteristic polynomials are identical (though their multiplicities may differ).
• [n*n]: The minimum polynomial of A is a factor of its characteristic polynomial.
• [n*n]: If A is nilpotent to index k its minimal polynomial is t^k.

Singular Values

[m*n, m>=n] The singular values of A are the positive square roots of the eigenvalues of A'A.

• [n*n]: If A is normal, its singular values and the absolute values of its eigenvalues.
• [n*n]: A is non-singular iff all its singular values are > 0.