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

- If
**A**is real, then**U**is unique and real. - We can also decompose
**A**as**L**^{H}**L**=**A**where**L**is lower triangular. If**V**^{H}**V**=**B**is the Cholesky decomposition of**B**=**JAJ**, then**L**^{H}**L**=**A**where**L**=**JVJ**. [Note:**J**is the exchange matrix.] **A**=**S**^{H}**S**iff**S**_{[m#n]}=**Q**[**U**;**0**] for some unitary**Q**.- For
**A**_{[n#n]}, the calculation of**U**requires*n*^{3}/6 flops and is numerically well conditioned.- A numerically and computationally good way to solve
**Ax**=**b**, is to calculate**U**, then solve**U**^{H}**y**=**b**and finally solve**Ux**=**y**.

- A numerically and computationally good way to solve

[**A**,**B**:*n#n*, hermitian] If
**B** is positive definite there exists **X** such that
**X**^{H}**BX**=**I** and
**X**^{H}**AX**=**D** where **X** and
**D** may be obtained from the eigendecomposition
**B**^{-1}**A**=**XDX**^{-1} with
**D**=**DIAG**(**d**) a diagonal matrix
of eigenvalues in non-increasing order. [4.10]

- If
**A**and**B**are real then so are**X**and**d**. [4.10] - If
**S**is the +ve definite hermitian square root of**B**^{-1}(i.e.**S**^{2}**B**=**I**), then the eigendecomposition of the hermitian matrix**SAS**is**SAS**=**WDW**^{H}where**W**=**S**^{-1}**X**is unitary. [4.10]

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**.

- The matrix
**C**is uniquely determined by**A**. - The number of non-zero rows of
**C**equals the rank of**A**.

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

**AB**=**BA**- rank(
**B**) = rank(**A**) **B**is +ve definite iff**A**is.**B**is real iff**A**is.**B**=p(**A**) for some polynomial p().

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

**B**is upper bidiagonal and its main diagonal consists of the eigenvalues of**A**repeated according to their algebraic multiplicities.**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).

- The matrix
**B**is unique except for the ordering of the blocks in**DIAG**(**X, Y, …**). - Two matrices are similar iff they
have the same Jordan form for some ordering of the blocks in
**DIAG**(**X, Y, …**). - The matrix
**B**may be complex even if**A**is real. - Each distinct eigenvalue of
**A**corresponds to one or more blocks in**B**that have that eigenvalue along their diagonal:- Its algebraic multiplicity equals the sum of the sizes of the blocks and equals its multiplicity in the characteristic polynomial of A.
- Its geometric multiplicity equals the number of blocks; this equals the number of distinct eigenvectors corresponding to the eigenvalue.
- Its multiplicity in the minimum polynomial equals the size of the largest block.

- The matrix
**B**is diagonal iff**A**is non-defective, i.e. the geometric and algebraic multiplicities are equal for each eigenvalue, i.e. each block is of size 1. - The minimum polynomial is equal to the characteristic polynomial iff all eigenvalues have a geometric multiplicity of 1, i.e. there is only one distinct eigenvector for each eigenvalue.
- Some authors call
**A***irreducible*iff**B**is a hypercompanion matrix (i.e. it consists of a single block) but a more common definition of*reducible*is here.

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

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.

- If
**A**is hermitian then**U**=**L**^{H}. - You can also decompose as
**A**=**PUDL**by expressing**JAJ**=(**JPJ**)(**JUJ**)(**JDJ**)(**JLJ**). [Note:**J**is the exchange matrix.]

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.

- This decomposition is the standard way of solving the simultaneous
equations
**A****x**=**b**. - You can also decompose as
**A**=**LUP**where**U**is unit upper triangular by expressing**A**^{T}=**P**^{T}**U**^{T}**L**^{T}or alternatively**A**^{H}=**P**^{H}**U**^{H}**L**^{H}. - You can also decompose as
**A**=**PUL**where**U**is unit upper triangular by expressing**JAJ**=(**JPJ**)(**JUJ**)(**JLJ**). [Note:**J**is the exchange matrix.]

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.

**H**^{2}=**A**^{H}**A**and**K**^{2}=**AA**^{H}**A**is normal iff**UH**=**HU**iff**KV**=**VK**. If**A**is normal and nonsingular then**U**=**V**and**H**=**K**.

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].

- For any
**A**_{[m#n]}with*m>=n*,**A=Q**_{[m#n]}**R**_{[n#n]}with**Q**^{H}**Q**=**I**and**R**upper triangular. If**A**is real then**Q**and**R**may be taken to be real [1.9]. - You can also decompose as
**A**=**RQ**by decomposing**JA**^{T}**J**=(**JQ**^{T}**J**)(**JR**^{T}**J**). [Note:**J**is the exchange matrix.] - You can also decompose as
**A**=**QL**where**L**is lower triangular by decomposing**JAJ**=(**JQJ**)(**JLJ**). - You can also decompose as
**A**=**LQ**where**L**is lower triangular by decomposing**A**^{T}=**Q**^{T}**L**^{T}.

Every square matrix **A** is unitarily similar to an upper triangular matrix **T** with
**A**=**U**^{H}**TU**.

- The main diagonal of
**T**contains the eigenvalues of**A**repeated according to their algebraic multiplicities.- The eigenvalues may be chosen to occur in any order along the diagonal of
**T**and for each possible order the matrix**U**is unique.

- The eigenvalues may be chosen to occur in any order along the diagonal of
**T**is diagonal iff**A**is normal. The sum of the squares of the absolute values of the off-diagonal elements of**T**is**A**'s*departure from normality*.

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

**T**is diagonal iff**A**is symmetric.

For **A**_{[n#n]} and **B**_{[n#n]},
there exist unitary **U** and **V**
and upper triangular **S** and
**T** such that **A**=**U**^{H}**SV** and
**B**=**U**^{H}**TV**.

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**=**U**^{T}**SV** and
**B**=**U**^{T}**TV** where each block of
**S** and **T** is either a 1#1 matrix or else a 2#2 matrix having
complex conjugate eigenvalues.

Every matrix **A**_{[m#n]} can be expressed as
**A**=**UDV**^{H} 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. *d _{i,i}*
<=

- If
**A**is real then**U**and**V**can be chosen to be real orthogonal matrices. - The matrix
**D**is unique but the matrices**U**and**V**are not. **A**=**LDM**^{H}is an alternative singular value decomposition of**A**iff**U**^{H}**L**=**DIAG**(**Q**_{1},**Q**_{2}, ...,**Q**_{k},**R**) and**V**^{H}**M**=**DIAG**(**Q**_{1},**Q**_{2}, ...,**Q**_{k},**S**) where**Q**_{1},**Q**_{2}, ...,**Q**_{k}are unitary matrices whose sizes are given by the multiplicities of the corresponding distinct non-zero singular values and**R**and**S**are unitary matrices whose size equals the number of zero singular values [1.14]- If the singular values are all distinct and non-zero then
**L**=**U Q**and**M**=**V Q**where**Q**is a diagonal matrix whose diagonal elements have unit magnitude. In the case of a real SVD of a real matrix**A**, the diagonal elements of**Q**equal +-1.

- If the singular values are all distinct and non-zero then
- The non-zero singular values of
**A**are the square roots of the non-zero eigenvalues of**A**^{H}**A**. **A**is non-singular iff all its singular values are > 0.- The rank of
**A**is the number of non-zero singular values. If the rank of**A**is*r*, then the first*r*columns of**U**form a basis for the range of**A**and the last*n-r*columns of**V**form a basis for the null space of**A**. - [
*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*]).

See also: Singular Value

Every symmetric (possibly complex) matrix **A** can be
expressed as **A**=**UDU**^{T} where
**U** is unitary and
**D** is real, non-negative and diagonal with its diagonal
elements arranged in non-increasing order (i.e. *d _{i,i}* <=

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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