Matrix Decompositions

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Cholesky Decomposition

Iff A is hermitian positive definite there exists a non-singular upper triangular U with positive real diagonal entries such that UHU=A . This is the Cholesky decomposition of A.


Dual Conjunctive Diagonalization

[A,B:n#n, hermitian] If B is positive definite there exists X such that XHBX=I and XHAX=DIAG(d) where d contains the eigenvalues of B-1A in non-increasing order and the columns of X the corresponding eigenvectors. [4.10]


Hermite Normal Form or Row-Echelon Form

Every matrix A[m#n] can be expressed as A=BCP where B[m#m] is non-singular, P[n#n] is a permutation matrix and C[m#n] is of the form [I D;0] for some D. The matrix C is the row-echelon or Hermite-normal form of A.


kth Root of a Hermitian Matrix

If A is hermitian +ve semidefinite, then for any integer k>0, there exists a unique +ve semidefinite B such that A=Bk.


Jordan Normal Form

For any A[n#n], there exists X such that A=X-1BX where B is of the following form:

  1. B is upper bidiagonal and its main diagonal consists of the eigenvalues of A repeated according to their algebraic multiplicities.
  2. B = DIAG(X,Y,…) where each square block X, Y, … has one of A's eigenvalues repeated along its main diagonal, 1's along the diagonal above it and 0's elsewhere (these blocks are hypercompanion matrices).

Finding the Jordan form of a defective or nearly defective matrix is numerically ill-conditioned.


LDU Decomposition

Every non-singular square matrix A can be expressed as A=PLDU where P is a permutation matrix, L is unit lower triangular, D is diagonal and U is unit upper triangular.


LU Decomposition

Every A[n#n] can be expressed as A=PLU where P is a permutation matrix, L is a unit lower triangular matrix and U is upper triangular.


Polar Factorisation

Every A[n#n] can be expressed as A=UH=KV with U, V unitary and H, K positive semidefinite hermitian. H and K are unique; U and V are unique iff A is nonsingular. If A is real then H and K are real and U and V may be taken to be real.


QR Decomposition

For any A[m#n], A=QR for Q[m#m] unitary and R[m#n] upper triangular. If A is real then Q and R may be taken to be real [1.9].


Schur Decomposition

Every square matrix A is unitarily similar to an upper triangular matrix T with A=UHTU.


Schur Decomposition, Real

Every square real matrix A is orthogonally similar to an upper block triangular matrix T with A=QTTQ where each block of T is either a 1#1 matrix or a 2#2 matrix having complex conjugate eigenvalues.


Schur Decomposition, Generalized

For A[n#n] and B[n#n], there exist unitary U and V and upper triangular S and T such that A=UHSV and B=UHTV.


Schur Decomposition, Generalized Real

If A[n#n] and B[n#n] are real, then there exist orthogonal U and V and upper block triangular S and T such that A=UTSV and B=UTTV  where each block of S and T is either a 1#1 matrix or else a 2#2 matrix having complex conjugate eigenvalues.


Singular Value Decomposition (SVD)

Every matrix A[m#n] can be expressed as A=UDVH where U[m#m] and V[n#n] are unitary and D[m#n] is real, non-negative and diagonal with its diagonal elements arranged in non-increasing order (i.e. di,i <= dj,j for i < j). The singular values of A are the diagonal elements of D (or to some people, the non-zero diagonal elements of D).

See also: Singular Value


Takagi Decomposition (SVD)

Every symmetric (possibly complex) matrix A can be expressed as A=UDUT where U is unitary and D is real, non-negative and diagonal with its diagonal elements arranged in non-increasing order (i.e. di,i <= dj,j for i < j). This is a special case of the singular value decomposition.


This page is part of The Matrix Reference Manual. Copyright © 1998-2017 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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