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单词 ConjugateTranspose
释义

conjugate transpose


Definition If A is a complex matrix, then theconjugate transposeMathworldPlanetmath A is the matrixA=A¯T, where A¯ isthe complex conjugateDlmfMathworldPlanetmath of A, and AT is thetransposeMathworldPlanetmath of A.

It is clear that for real matrices, the conjugate transpose coincides withthe transpose.

0.0.1 Properties

  1. 1.

    If A and B are complex matrices of same size, and α,βare complex constants, then

    (αA+βB)=α¯A+β¯B,
    A=A.
  2. 2.

    If A and B are complex matrices such that AB is defined, then

    (AB)=BA.
  3. 3.

    If A is a complex square matrixMathworldPlanetmath, then

    det(A)=detA¯,
    trace(A)=traceA¯,
    (A)-1=(A-1),

    where trace and det are the traceand the determinantDlmfMathworldPlanetmath operators, and -1 is the inverse operator.

  4. 4.

    Suppose , is the standard inner product on n.Then for an arbitrary complex n×n matrix A,and vectors x,yn, we have

    Ax,y=x,Ay.

Notes

The conjugate transpose of A is also called the adjoint matrix of A,the Hermitian conjugate of A (whence one usually writes A=AH).The notation A is also used for the conjugate transpose [2].In [1], A is also called the tranjugate of A.

References

  • 1 H. Eves, Elementary MatrixMathworldPlanetmath Theory, Dover publications, 1980.
  • 2 M. C. Pease,Methods of Matrix Algebra, Academic Press, 1965.

See also

  • Wikipedia,http://www.wikipedia.org/wiki/Conjugate_transposeconjugate transpose

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更新时间:2025/5/4 9:25:12