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单词 Transpose
释义

transpose


The transposeMathworldPlanetmath of a matrix A is the matrix formed by “flipping” A about the diagonal line from the upper left corner. It is usually denoted At, although sometimes it is written as AT or A. So if A is an m×n matrix and

A=(a11a12a1na21a22a2nam1am2amn)

then

At=(a11a21am1a12a22am2a1na2nanm)

Note that the transpose of an m×n matrix is a n×m matrix.

Properties

Let A and B be m×m matrices, C and D be m×n matrices, E be an n×k matrix, and c be a constant. Let x and y be column vectorsMathworldPlanetmath with n rows. Then

  1. 1.

    (Ct)t=C

  2. 2.

    (C+D)t=Ct+Dt

  3. 3.

    (cD)t=cDt

  4. 4.

    (DE)t=EtDt.

  5. 5.

    (AB)t=BtAt.

  6. 6.

    If A is invertiblePlanetmathPlanetmath , then (At)-1=(A-1)t

  7. 7.

    If A is real, trace(AtA)0 (where trace is the trace of a matrix).

  8. 8.

    The transpose is a linear mapping from the vector spaceMathworldPlanetmath of matrices to itself. That is, (αA+βB)t=α(A)t+β(B)t, for same-sized matrices A and B and scalars α and β.

The familiar vector dot productMathworldPlanetmath can also be defined using the matrix transpose. If x and y are column vectors with n rows each,

xty=xy

which implies

xtx=xx=||x||22

which is another way of defining the square of the vector Euclidean norm.

Titletranspose
Canonical nameTranspose
Date of creation2013-03-22 12:01:02
Last modified on2013-03-22 12:01:02
Ownermathcam (2727)
Last modified bymathcam (2727)
Numerical id12
Authormathcam (2727)
Entry typeDefinition
Classificationmsc 15A57
Related topicAdjointEndomorphism
Related topicHermitianConjugate
Related topicFrobeniusMatrixNorm
Related topicConjugateTranspose
Related topicTransposeOperator
Related topicVectorizationOfMatrix
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更新时间:2025/5/4 14:37:29