Transpose (original) (raw)

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 _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 vectorss expressed as columns of their coordinates can be computed as

a · b = aT b

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_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 = _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 skew-symmetric, i.e. A is skew-symmetric 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.

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

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.