Transpose

The transpose of a matrix A, denoted A^T, is an operation that interchanges the corresponding rows and columns of a matrix:

Specifically, transposing a matrix is an operation that changes the position of each index in a matrix (refer to the matrix notation page for a refresher if necessary), such that an n x m matrix becomes an m by n matrix, making the (i,j)^th entry of the matrix the (j,i)^th. For example, in matrix A above, entry a₁₁, becomes entry a₂₁, which in both cases is 1. Similarly, entry a₂₃ becomes entry a₃₂, such that the "6" is transposed from the 3rd column of the 2nd row, to the 2nd column of the 3rd row in the transposed matrix. The general form of a matrix and its transpose can be written as follows:

Note that in the transposed matrix A^T, the indices no longer indicate the corresponding entry in the matrix. Rather, it shows that the first row in matrix A becomes the first column in the transposed matrix.

Properties of transposition

Below is a list of properties of matrices that involve transposition, as well as examples of each:

1. 
2. , for some scalar k
3. 
4. 
5. , where A is a square matrix
6. , where A is a square invertible matrix

i. For ,

;

iii. For and

iv. For For and ,

Thus:

v. For , where "det" indicates taking the determinant of the matrix,

Thus:

vi. For , where A^-1 indicates the inverse of matrix A,

Thus: