Matrix Transposition

Introduction

The transpose of a matrix is a flipped version.

┌    ┐    ┌    ┐
│ 1 2│    │ 1 3│
│ 3 4│ -> │ 2 4│
└    ┘    └    ┘

The transpose of A is denoted A^T.

Multiplication of Transposed Matrices

The transpose of a product is the same as the reversed product of the transposed multiples. (A B)^T = B^T A^T.

Inverses of Transposed Matrices

A simple proof based on the definition of inverse matrices and the above multiplicative property:

      -1        -1
   A A   = I = A   A

(leave the left side off for now)

      -1
   A A   = I

       T
     -1     T
  A A    = I

       T
     -1
  A A    = I

   T
 -1   T
A    A   = I

(bring back the left side)

   T
 -1   T        -1
A    A   = I = A   A

(and it should now be clear that)

   T     -1
 -1     T
A    = A

Inverses and transposes can be done in any order.

CategoryRicottone