Anti-Hermitian matrix

In mathematics, an anti-Hermitian or skew-Hermitian matrix is a square matrix A , whose Hermitian conjugation changes the sign of the original matrix:

A^{\dagger }=-A,

{\ displaystyle A ^ {\ dagger} = - A,}

{\ displaystyle A ^ {\ dagger} = - A,}

or element by element:

a_{i,j}=-{\overline {a_{j,i}}},

{\ displaystyle a_ {i, j} = - {\ overline {a_ {j, i}}},}

{\ displaystyle a_ {i, j} = - {\ overline {a_ {j, i}}},}

where through ${\overline {x}}$ ${\ displaystyle {\ overline {x}}}$ $\ overline {x}$ the complex conjugation of a number is indicated $x$ ${\ displaystyle x}$ $x$ .

Properties

A matrix B is Hermitian if and only if the matrix i B is anti-Hermitian. It follows that if A is anti-Hermitian, then the matrices ± iA are Hermitian. Also, any anti-Hermitian matrix A can be represented in the form A = i B , where B is Hermitian. Thus, the properties of anti-Hermitian matrices can be expressed using the properties of Hermitian matrices and vice versa.
The matrix A is anti-Hermitian if and only if $X^{\dagger }A^{\dagger }Y=-X^{\dagger }AY$ ${\ displaystyle X ^ {\ dagger} A ^ {\ dagger} Y = -X ^ {\ dagger} AY}$ for any vectors $X$ ${\ displaystyle X}$ and $Y$ ${\ displaystyle Y}$ (the form $X^{\dagger }AY$ ${\ displaystyle X ^ {\ dagger} AY}$ - anti-Hermitian).
Anti-Hermitian matrices are closed with respect to addition, multiplication by a real number, raising to an odd degree, inversion (non-degenerate matrices).
Anti-Hermitian matrices are normal .
The even degree of the anti-Hermitian matrix is a Hermitian matrix. In particular, if $A$ ${\ displaystyle A}$ anti-Hermitian then $A^{2}$ ${\ displaystyle A ^ {2}}$ Hermitian.
The eigenvalues of the anti-Hermitian matrix are either zero or purely imaginary .
Any square matrix can be represented as the sum of Hermitian and anti-Hermitian:

M=M_{h}+M_{a}

{\ displaystyle M = M_ {h} + M_ {a}}

,

Where

M_{h}={\frac {1}{2}}(M+M^{\dagger })

{\ displaystyle M_ {h} = {\ frac {1} {2}} (M + M ^ {\ dagger})}

- Hermitian,

M_{a}={\frac {1}{2}}(M-M^{\dagger })

{\ displaystyle M_ {a} = {\ frac {1} {2}} (MM ^ {\ dagger})}

- anti-Hermitian.

Matrix $A$ ${\ displaystyle A}$ anti-Hermitian if and only if its exhibitor $e^{A}$ ${\ displaystyle e ^ {A}}$ unitary .

Anti-Hermitian matrices form a Lie algebra ${\mathfrak {u}}(n)$ ${\ displaystyle {\ mathfrak {u}} (n)}$ Lee groups $U(n)$ ${\ displaystyle U (n)}$ .

For any complex number $\lambda$ ${\ displaystyle \ lambda}$ such that $|\lambda |=1$ ${\ displaystyle | \ lambda | = 1}$ , there is a one-to-one correspondence between unitary matrices $U$ ${\ displaystyle U}$ having no eigenvalues equal $a$ ${\ displaystyle a}$ , and anti-Hermitian matrices $A$ ${\ displaystyle A}$ defined by Cayley's formulas:

U=\lambda (A-I)(A+I)^{-1},

{\ displaystyle U = \ lambda (AI) (A + I) ^ {- 1},}

A=\lambda (aI+U)(aI-U)^{-1},

{\ displaystyle A = \ lambda (aI + U) (aI-U) ^ {- 1},}

Where

I

{\ displaystyle I}

Is the identity matrix .

In particular, when

\lambda =-1

{\ displaystyle \ lambda = -1}

:

U=(I-A)(I+A)^{-1},

{\ displaystyle U = (IA) (I + A) ^ {- 1},}

A=(I-U)(I+U)^{-1}.

{\ displaystyle A = (IU) (I + U) ^ {- 1}.}

Links

Brookes, M., "The Matrix Reference Manual", Imperial College, London, UK

Anti-Hermitian matrix

Properties

See also

Links

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