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⇤ ← Revision 1 as of 2022-03-16 04:04:09
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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 AT.
Multiplication of Transposed Matrices
The transpose of a product is the same as the reversed product of the transposed multiples. (A B)T = BT AT.
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 = AInverses and transposes can be done in any order.