skew-symmetric matrix
Definition:
Let be an square matrix of
order with real entries .
The matrix is skew-symmetric if for all .
The main diagonal entries are zero because implies .
One can see skew-symmetric matrices as a special case of complex skew-Hermitian matrices. Thus, all properties of skew-Hermitian matrices also hold for skew-symmetric matrices.
Properties:
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1.
The matrix is skew-symmetric if and only if , where is the matrix transpose
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2.
For the trace operator, we have that . Combining this with property (1), it follows that for a skew-symmetric matrix .
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3.
Skew-symmetric matrices form a vector space: If and are skew-symmetric and , then is also skew-symmetric.
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4.
Suppose is a skew-symmetric matrix and is a matrix of same order as . Then is skew-symmetric.
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5.
All eigenvalues of skew-symmetric matrices are purely imaginary or zero. This result is proven on the page for skew-Hermitian matrices.
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6.
According to Jacobi’s Theorem, the determinant of a skew-symmetric matrix of odd order is zero.
Examples:
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Title | skew-symmetric matrix |
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Canonical name | SkewsymmetricMatrix |
Date of creation | 2013-03-22 12:01:05 |
Last modified on | 2013-03-22 12:01:05 |
Owner | Daume (40) |
Last modified by | Daume (40) |
Numerical id | 10 |
Author | Daume (40) |
Entry type | Definition |
Classification | msc 15-00 |
Related topic | SelfDual |
Related topic | AntiSymmetric |
Related topic | SkewHermitianMatrix |
Related topic | AntisymmetricMapping |