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, ^tA, A', or A^T, the latter notation being preferred in Wikipedia.

Formally, the transpose of the m-by-n matrix A is the n-by-m 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 m-by-n 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 m-by-n matrices to the space of all n-by-m matrices.

The transpose operation is self-inverse, i.e taking the transpose of the transpose amounts to doing nothing: (A^T)^T = A.

If A is an m-by-n and B an n-by-k 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

where the product on the right is the ordinary matrix multiplication.

If A is an arbitrary m-by-n matrix with real entries, then A^TA is a positive semidefinite matrix.

If A is an n-by-n matrix over some field, then A is similar to A^T.

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:

    the identity matrix

A square matrix whose transpose is equal to its negative is called skew-symmetric, i.e. A is skew-symmetric 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 ^tf : W* -> V* with

    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.