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transpose (Definition)

The transpose of a matrix $ A$ is the matrix formed by “flipping” $ A$ about the diagonal line from the upper left corner. It is usually denoted $ A^t$, although sometimes it is written as $ A^T$ or $ A'$. So if $ A$ is an $ m \times n$ matrix and

$\displaystyle A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ a_{21} ... ... & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}$
then

$\displaystyle A^t = \begin{pmatrix} a_{11} & a_{21} & \cdots & a_{m1} \ a_{12... ... & \vdots & \ddots & \vdots \ a_{1n} & a_{2n} & \cdots & a_{nm} \end{pmatrix}$

Note that the transpose of an $ m \times n$ matrix is a $ n \times m$ matrix.

Properties

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

  1. $ (C^t)^t = C$
  2. $ (C+D)^t = C^t + D^t$
  3. $ (cD)^t = cD^t$
  4. $ (DE)^t=E^tD^t.$
  5. $ (AB)^t = B^t A^t.$
  6. If $ A$ is invertible , then $ (A^t)^{-1} = (A^{-1})^t $
  7. If $ A$ is real, $ \operatorname{trace}(A^tA) \ge 0$ (where $ \operatorname{trace}$ is the trace of a matrix).
  8. The transpose is a linear mapping from the vector space of matrices to itself. That is, $ (\alpha A + \beta B)^t = \alpha (A)^t + \beta (B)^t$, for same-sized matrices $ A$ and $ B$ and scalars $ \alpha$ and $ \beta$.

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

$\displaystyle x^t y = x \cdot y $

which implies

$\displaystyle x^t x = x \cdot x = \vert\vert x\vert\vert _2^2 $

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



"transpose" is owned by mathcam. [ full author list (2) | owner history (1) ]
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See Also: adjoint endomorphism, Hermitian conjugate, Frobenius matrix norm, conjugate transpose, transpose operator, vectorization of matrix
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Cross-references: vector Euclidean norm, square, implies, dot product, vector, scalars, vector space, linear mapping, trace of a matrix, real, invertible, rows, column vectors, line, diagonal, matrix
There are 49 references to this entry.

This is version 8 of transpose, born on 2001-11-20, modified 2005-03-18.
Object id is 975, canonical name is Transpose.
Accessed 23178 times total.

Classification:
AMS MSC: 15A57 (Linear and multilinear algebra; matrix theory :: Other types of matrices )
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