Hermitian and Skew-Hermitian Matrices: Amatrixais Said to Be Hermitian If A∗ = A, and It Is Called Skew-Hermitian If A∗ = −A

Home , Hermitian matrix

• Hermitian and Skew-Hermitian Matrices: AmatrixAis said to be Hermitian if A∗ = A, and it is called Skew-Hermitian if A∗ = −A.

• Rayleigh Quotient: The Rayleigh Quotient associated to an n × n matrix A is deﬁned as follows: hAx,xi R (x):= , x∈Cn\{0}. A hx,xi

• Positive Semidefinite and Positive Definite Matrices: AmatrixA∈Mn is said to be positive semidefinite if A is Hermitian and hAx, xi≥0 for every x ∈ Cn.WesaythatAis positive definite if A is Hermitian and hAx, xi > 0 for every x ∈ Cn \{0}.

• Jordan Blocks and Jordan Matrices: Let λ ∈ C and let k be a positive integer. The Jordan Block of size k associated to λ is the k × k matrix Jk(λ) deﬁned as follows: J1(λ):=[λ]; for k ≥ 2, the entries of Jk(λ)aregivenby ( λ, if m = l, 1 ≤ l ≤ k; [Jk(λ)]lm := 1, if m = l +1, 1≤l≤k−1; 0, otherwise.

A direct sum of Jordan blocks is called a Jordan Matrix.

• Rayleigh–Ritz Min-Max Principle: Let λ1 ≤ ··· ≤ λn be the eigenvalues of an n × n Hermitian matrix A.Let{v1,...,vn} be an orthonormal set of eigenvectors, with Avk = λkvk, ≤ k ≤ n R A 1 .Let A be the Rayleigh quotient associated to . Deﬁne

Vk := span{v1,...,vk} and Wk := span{vk,...,vn}, 1≤k ≤n.

The following hold for every 1 ≤ k ≤ n: λ {R ∈V \{ }} (i) k =max A (x):x k 0 . R λ ∈V \{ } λ (ii) A (x)= k,x k 0 , if and only if x is an eigenvector associated to k. λ {R ∈W \{ }} (iii) k =min A (x):x k 0 . R λ ∈W \{ } λ (iv) A (x)= k,x k 0 , if and only if x is an eigenvector associated to k.

• Courant–Fischer Theorem: Let λ1 ≤ ··· ≤ λn be the eigenvalues of an n × n Hermitian A R A matrix ,andlet A be the Rayleigh quotient associated to . The following hold for every 1 ≤ r ≤ n − 1: λ { {R 6 , ⊥{ ,..., }}} (i) n−r =min max A (x):x=0 x c1 cr , where the outer minimum is taken over n all possible sets {c1,...,cr} of r vectors in C . λ { {R 6 , ⊥{ ,..., }}} (ii) r+1 =max min A (x):x=0 x c1 cr , where the outer maximum is taken over n all possible sets {c1,...,cr} of r vectors in C . Yn • Hadamard’s Inequality: If A ∈ Mn is positive deﬁnite, then det(A) ≤ [A]kk, with equality k=1 holding if and only if A is diagonal.

• Ostrowski–Taussky Inequality: Let A be an n × n matrix whose Hermitian part A+A∗ H(A):= is positive deﬁnite. Then det(H(A)) ≤|det(A)|, with equality holding if and only 2 if A is Hermitian.

1 • Jordan’s Theorem: Suppose that A ∈ Mn. There exists an invertible matrix S and a Jordan matrix J such that S−1AS = J.IfAis a real matrix and all its eigenvalues are real, then S can be chosen to be real. Furthermore, if α is an eigenvalue of A, then its algebraic multiplicity is the number of times α appears on the diagonal of J, whilst its geometric multiplicity is the number of Jordan blocks in J in which α appears as the common diagonal entry. The matrix J is unique up to permutation of its Jordan blocks.

• Equivalent conditions for positive deﬁniteness: Suppose that A is an n × n Hermitian matrix. The following are equivalent: (a) A is positive deﬁnite. (b) All the eigenvalues of A are positive. (c) For every 1 ≤ r ≤ n, the determinant of the r × r principal submatrix of A is positive. (d) A can be reduced, via Gaussian elimination using only type-three operations, to an upper- triangular matrix all of whose diagonal entries are positive. (e) A can be factored in the form A = LT ,whereLis a unit-lower-triangular matrix and T is an upper-triangular matrix all of whose diagonal entries are positive. (f) A canfactoredintheformA=LDL∗,whereLis unit lower triangular and D is a diagonal matrix all of whose diagonal entries are positive. (g) A admits a Cholesky factorization, namely A = CC∗,whereCis a lower-triangular matrix all of whose diagonal entries are positive. (h) A can be factored in the form A = B∗B where B is nonsingular.