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Nomal matrix

$Adjoint matrix$

Adjoint matrix of

A

:

A^{*} = (\overline{a}_{ji})

$Normal matrix$

A

is a normal matrix:

A A^{*} = A^{*} A

$Projection$

Projection from

V

to

W

:

A : V \to W (x \mapsto x_{1})

$Symmetric, alternating matrix$

A

is a symmetric matrix:

A = A^{T}

A

is an alternating matrix:

A = - A^{T}

$Hermitian matrix$

A

is a Hermitian matrix :

A = A^{*}

$Unitary matrix$

U

is a unitary matrix:

U^{*} U = E

$Corollary$

U V

is a unitary matrix

$Corollary$

U^{- 1} = U^{*}

$Corollary$

Unitary matrices are normal matrices

$Corollary$

U^{*}

is a unitary matrix

$Corollary$

set of all the unitary matrices are group

$Corollary$

∣ det U ∣ = 1

$Corollary$

∣ λ ∣ = 1

$Corollary$

U^{*} A^{n} U = (U^{*} A U)^{n}

$Corollary$

以下同値

1. 行列 A はユニタリ行列 2. ⟨ U x, U y ⟩ = ⟨ x, y ⟩

$Corollary$

以下同値

1. 行列 U はユニタリ行列 2.∥ U x ∥ = ∥ x ∥

$Corollary$

Column vectors of unitary matrix are orthonormal basis

$Corollary$

U^{- 1} A U = A