Matrix Transposition

Matrix transposition is the interchange of matrix rows and columns.

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Remark

In the square case, $A \to A^{T}$ is "reflecting" the matrix entries across the main diagonal.

Definition

Let $A$ be a $m \times n$ matrix, then its transpose, denoted $A^{T}$ , is the $n \times m$ matrix with ${(A^{T})}_{i j} = A_{j i}$ .

Basic Properties:

{(A^{T})}^{T} = A

(A + B)^{T} = A^{T} + B^{T}

(α A)^{T} = α A^{T}

(A B)^{T} = B^{T} A^{T}

Inversion and transposition respect each other:

Theorem

If $A$ is a nonsingular matrix so is $A^{T}$ , and $(A^{T})^{- 1} = (A^{- 1})^{T}$ .

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