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, 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,
Properties 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 mbyn matrices to the space of all nbym 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 vectors expressed as columns of their coordinates can be computed as
If A is an arbitrary mbyn matrix with real entries, then A^{T}A is a positive semidefinite matrix.
Further nomenclature A square matrix whose transpose is equal to itself is called a symmetric matrix, i.e. A is symmetric iff
A square matrix whose transpose is also its inverse is called an orthogonal matrix, i.e. G is orthogonal iff:
A square matrix whose transpose is equal to its negative is called skewsymmetric, i.e. A is skewsymmetric iff
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.
Transpose of linear maps 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
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.
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