Transpose
See also Transposition for meanings of this term in telecommunication and music.
In mathematics, and in particular linear algebra, the transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. Informally, the transpose of a square matrix is obtained by reflecting at the main diagonal (that runs from the top left to bottom right of the matrix).
The transpose of the matrix A is written as A^{tr}, ^{t}A, A', or A^{T}, the latter notation being preferred in Wikipedia.
Formally, the transpose of the mbyn matrix A is the nbym matrix A^{T} defined by
A^{T}[i, j] = A[j, i] for 1 ≤ i ≤ n and 1 ≤ j ≤ m.
For example,

For any two mbyn matrices A and B and every scalar c, we have (
A + B)
^{T} =
A^{T} +
B^{T} and (
cA)
^{T} =
c(
A^{T}). This shows that the transpose is a
linear map from the space of all
mby
n matrices to the space of all
nby
m matrices.
The transpose operation is selfinverse, i.e taking the transpose of the transpose amounts to doing nothing: (A^{T})^{T} = A.
If A is an mbyn and B an nbyk matrix, then we have (AB)^{T} = (B^{T})(A^{T}). Note that the order of the factors switches. 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}.
The dot product of two vectorss expressed as columns of their coordinates can be computed as
 a · b = a^{T} b
where the product on the right is the ordinary matrix multiplication.
If A is an arbitrary mbyn matrix with real entries, then A^{T}A is a positive semidefinite matrix.
A square matrix whose transpose is equal to itself is called a symmetric matrix, i.e.
A is symmetric
iff
 A = A^{T}
A square matrix whose transpose is also its inverse is called an
orthogonal matrix, i.e.
G is orthogonal iff:
 G G^{T} = G^{T} G = I_{n}
A square matrix whose transpose is equal to its negative is called
skewsymmetric, i.e.
A is skewsymmetric iff
 A =  A^{T}
The
conjugate transpose of the complex matrix
A, written as
A^{*}, is obtained by taking the transpose of
A and then taking the
complex conjugate of each entry.
If f: V > W is a linear map between vector spaces V and W with dual spaces W* and V*, we define the
transpose of
f to be the linear map
^{t}f : W*
> V* with
 ^{t}f (φ) = φ o f for every φ in W*.
If the matrix
A describes a linear map with respect to two
bases, then the matrix
A^{T} describes the transpose of that linear map with respect to the dual bases. See
dual space for more details on this.