Skew-symmetric matrix

Antisymmetric ( skew-symmetric or skew- symmetric ) matrix - square matrix $A$ ${\ displaystyle A}$ $A$ over the field $k$ ${\ displaystyle k}$ $k$ characteristics other than 2, satisfying the ratio:

A^{T}=-A,

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

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

Where $A^{T}$ ${\ displaystyle A ^ {T}}$ $A ^ {T}$ - transposed matrix .

For $n\times n$ ${\ displaystyle n \ times n}$ $n \ times n$ matrices $A$ ${\ displaystyle A}$ $A$ this ratio is equivalent to:

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

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

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

for all

i,j=1,2,\ldots ,n

{\ displaystyle i, j = 1,2, \ ldots, n}

{\ displaystyle i, j = 1,2, \ ldots, n}

,

Where $a_{i,j}$ ${\ displaystyle a_ {i, j}}$ $a_ {i, j}$ - element $i$ ${\ displaystyle i}$ $i$ th row and $j$ ${\ displaystyle j}$ $j$ matrix column $A$ ${\ displaystyle A}$ $A$ .

Properties

The rank of the skew-symmetric matrix is always even .
Any square matrix B over a characteristic field other than 2 is the sum of the symmetric and skew-symmetric matrices, which are uniquely determined.
Nonzero roots of the characteristic polynomial of a real skew-symmetric matrix are purely imaginary numbers .
A real skew-symmetric matrix is similar to a block-diagonal matrix with zero diagonal blocks and diagonal blocks $2\times 2$ ${\ displaystyle 2 \ times 2}$ kind of

{\begin{pmatrix}0&a\\-a&0\end{pmatrix}}

{\ displaystyle {\ begin {pmatrix} 0 & a \\ - a & 0 \ end {pmatrix}}}

.

The set of all skew-symmetric matrices of order $n$ ${\ displaystyle n}$ over the field $k$ ${\ displaystyle k}$ forms a Lie algebra over $k$ ${\ displaystyle k}$ regarding matrix addition and commutation:

[A,B]=AB-BA

{\ displaystyle [A, B] = AB-BA}

.

Skew-symmetric matrix

Properties

See also

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