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skew-Hermitian matrix
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(Definition)
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Definition. A square matrix with complex entries is skew-Hermitian, if
Here
,
is the transpose of , and
is is the complex conjugate of the matrix .
- The trace of a skew-Hermitian matrix is imaginary.
- The eigenvalues of a skew-Hermitian matrix are imaginary.
Proof. Property (1) follows directly from property (2) since the trace is the sum of the eigenvalues. But one can also give a simple proof as follows. Let and be the real respectively imaginary parts of the elements in . Then the diagonal elements of are of the form
, and the diagonal elements in are of the form
. Hence , i.e., the real part for the diagonal elements in must vanish, and property (1) follows. For property (2), suppose is a skew-Hermitian matrix, and an eigenvector corresponding to the eigenvalue , i.e.,
Here, is a complex column vector. Since is an eigenvector, is not the zero vector, and
. Without loss of generality we can assume
. Thus
Hence the eigenvalue corresponding to is imaginary.
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Cross-references: without loss of generality, zero vector, column vector, eigenvalue, eigenvector, vanish, real part, diagonal, imaginary parts, real, simple, sum, property, proof, eigenvalues, trace, matrix, complex conjugate, transpose, complex, square matrix
There are 5 references to this entry.
This is version 18 of skew-Hermitian matrix, born on 2003-04-29, modified 2006-06-14.
Object id is 4231, canonical name is SkewHermitianMatrix.
Accessed 14331 times total.
Classification:
| AMS MSC: | 15A57 (Linear and multilinear algebra; matrix theory :: Other types of matrices ) |
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Pending Errata and Addenda
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