Transpose - Properties

Properties

For matrices A, B and scalar c we have the following properties of transpose:

1. Taking the transpose is an involution (self-inverse).

2. The transpose respects addition.

3. Note that the order of the factors reverses. From this one can deduce that a square matrix A is invertible if and only if AT is invertible, and in this case we have (A−1)T = (AT)−1. By induction this result extends to the general case of multiple matrices, where we find that (A₁A₂...A_k-1A_k)T = A_kTA_k-1T... A₂TA₁T.

4. The transpose of a scalar is the same scalar. Together with (2), this states that the transpose is a linear map from the space of m × n matrices to the space of all n × m matrices.

5. The determinant of a square matrix is the same as that of its transpose.

6. The dot product of two column vectors a and b can be computed as
   which is written as a_i bi in Einstein notation.
7. If A has only real entries, then ATA is a positive-semidefinite matrix.

8. The transpose of an invertible matrix is also invertible, and its inverse is the transpose of the inverse of the original matrix. The notation A−T is often used to represent either of these equivalent expressions.

9. If A is a square matrix, then its eigenvalues are equal to the eigenvalues of its transpose.