Transpose

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See transposition for meanings of this term in telecommunication and music.

In linear algebra, the transpose of a matrix A is another matrix A^T (also written A^tr, ^tA, or A′) created by any one of the following equivalent actions:

write the rows of A as the columns of A^T
write the columns of A as the rows of A^T
reflect A by its main diagonal (which starts from the top left) to obtain A^T

Formally, the transpose of an m × n matrix A is the n × m matrix

$A^\mathrm{T}[i, j] = A[j, i] \quad \forall\ 1 < i < n, 1 < j < m.$

1 Examples
2 Properties
- 2.1 Proof
3 Special transpose matrices
4 Transpose of linear maps
5 External links

[edit] Examples

$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}^{\mathrm{T}} \!\! \;\! = \, \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}\quad\quad \mbox{and}\quad\quad \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}^{\mathrm{T}} \!\! \;\! = \, \begin{bmatrix} 1 & 3 & 5\\ 2 & 4 & 6 \end{bmatrix} \;$

[edit] Properties

For matrices A, B and scalar c we have the following properties of transpose:

$\left( A^\mathrm{T} \right)^\mathrm{T} = A \quad \,$

Taking the transpose is a self inverse.
$(A+B)^\mathrm{T} = A^\mathrm{T} + B^\mathrm{T} \,$

The transpose is a linear map from the space of m × n matrices to the space of all n × m matrices.
$\left( A B \right)^\mathrm{T} = B^\mathrm{T} A^\mathrm{T} \,$

Note that the order of the factors reverses. From this one can deduce that a square matrix A is invertible if and only if A^T is invertible, and in this case we have (A⁻¹)^T = (A^T)⁻¹.
$\left(c A\right)^\mathrm{T} = c A^\mathrm{T} \,$

The transpose of a scalar is the same scalar.
The dot product of two column vectors a and b can be computed as

$\mathbf{a} \cdot \mathbf{b} = \mathbf{a}^{\mathrm{T}} \mathbf{b},$

which is $a_i \, b_i$ in indicial notation.
If A has only real entries, then A^TA is a positive-semidefinite matrix.
If A is over some field, then A is similar to A^T.

[edit] Proof

To prove (AB)^T = B^T A^T: Suppose A is m × n and B is n × p (for the matrix multiplication to work). Consider the (i, j) element of the matrices at both sides of the equation. The (i, j) element of (AB)^T is the (j, i) element of AB by the definition of transpose. By the definition of matrix multiplication, this is

$\sum_{k=1}^n a_{jk} b_{ki}.$

The (i, j) element of B^T A^T is also

$\sum_{k=1}^n [B^\mathrm{T}]_{ik} \, [A^\mathrm{T}]_{kj} = \sum_{k=1}^n b_{ki} a_{jk}.$

[edit] Special transpose matrices

A square matrix whose transpose is equal to itself is called a symmetric matrix; that is, A is symmetric if

$A^{\mathrm{T}} = A.\,$

A square matrix whose transpose is also its inverse is called an orthogonal matrix; that is, G is orthogonal if

$G G^\mathrm{T} = G^\mathrm{T} G = I_n , \,$ the identity matrix.

A square matrix whose transpose is equal to its negative is called skew-symmetric; that is, A is skew-symmetric if

$A^{\mathrm{T}} = -A.\,$

The conjugate transpose of the complex matrix A, written as A^*, is obtained by taking the transpose of A and the complex conjugate of each entry:

$A^* = (\overline{A})^{-1} = \overline{(A^{-1})}$

[edit] Transpose of linear maps

If f: V→W is a linear map between vector spaces V and W with nondegenerate bilinear forms, we define the transpose of f to be the linear map ^tf : W→V, determined by

$B_V(v,{}^tf(w))=B_W(f(v),w) \quad \forall\ v \in V, w \in W.$

Here, B_V and B_W are the bilinear forms on V and W respectively. The matrix of the transpose of a map is the transposed matrix only if the bases are orthonormal with respect to their bilinear forms.

Over a complex vector space, one often works with sesquilinear forms instead of bilinear (conjugate-linear in one argument). The transpose of a map between such spaces is defined similarly, and the matrix of the transpose map is given by the conjugate transpose matrix if the bases are orthonormal. In this case, the transpose is also called the Hermitian adjoint.