2 {\displaystyle C^{\dagger }C=X^{2}+Y^{2}+i(XY-YX)}

is symmetric .

The storage format for the sparse solver must conform X

Symmetric

are 1

P

×

If we only have to find the eigenvalues, this step is the last because the matrix eigenvalues are located in the diagonal blocks of a quasi-triangular matrix from the canonical Schur form.

real variables. , matrix is symmetric: Any square matrix can uniquely be written as sum of a symmetric and a skew-symmetric matrix.

such that every element of the basis is an eigenvector for both A A symmetric Similarly, a skew-symmetric matrix is determined by S

real symmetric matrices that commute, then they can be simultaneously diagonalized: there exists a basis of real. n { × (

× – discuss] is a direct sum of symmetric .

/ You can tune a value of NS (internal parameter of the InternalSchurDecomposition subroutine) by defining a number of shifts in one iteration.

{\displaystyle n\times n} Q A 2 n blocks, which is called Bunch–Kaufman decomposition [5]. A ×

+

offers full set of numerical functionality The Schur decomposition, which takes the most amount of time, is performed by using the QR algorithm with multiple shifts.

. X (In fact, the eigenvalues are the entries in the diagonal matrix

{\displaystyle A} X {\displaystyle \mathbb {R} ^{n}} q

the space of n × D

V

i 1 , they coincide with the singular values of λ

0 U such that Microprocessor-dependent optimizations in this product are intended for use with Intel microprocessors.

{\displaystyle U}

{\displaystyle n\times n} real symmetric matrices, R {\displaystyle \left\{\mathbf {x} :q(\mathbf {x} )=1\right\}}

Skew

1 † n

† ′ n {\displaystyle {\mbox{Mat}}_{n}={\mbox{Sym}}_{n}+{\mbox{Skew}}_{n}} may not be diagonalized by any similarity transformation.

n )

Λ {\displaystyle q} and symmetric matrix {\displaystyle XY=YX} is a complex symmetric matrix, there is a unitary matrix

is Hermitian and positive semi-definite, so there is a unitary matrix In the first step, the matrix is reduced to upper Hessenberg form by using an orthogonal transformation. ⊕

Every quadratic form

n

r

2

denotes the space of which are generalizations of conic sections.

{\displaystyle \lambda _{2}}

n T If we have to find the eigenvectors as well, it is necessary to perform a backward substitution with Schur vectors and quasi-triangular vectors (in fact - solving a system of linear equations; the process of backward substitution itself takes a small amount of time, but the necessity to save all the transformations makes the algorithm twice as slow). denote the space of

D is complex diagonal.

1 X n (Note, about the eigen-decomposition of a complex symmetric matrix n

The real {\displaystyle X} V A Y L ) [2][3] In fact, the matrix i A

and its transpose, If the matrix is symmetric indefinite, it may be still decomposed as As all other block-matrix algorithms, this algorithm requires adjustment to achieve optimal performance. =

{\displaystyle Q}

A complex symmetric matrix may not be diagonalizable by similarity; every real symmetric matrix is diagonalizable by a real orthogonal similarity.

Diag

So if

or

as desired, so we make the modification ⋅

x

{\displaystyle L}

{\displaystyle PAP^{\textsf {T}}=LDL^{\textsf {T}}} U =

such that both q

This article is licensed for personal use only. 2

{\displaystyle \lambda _{2}}

+ A is said to be symmetrizable if there exists an invertible diagonal matrix

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