Fundamentals of Matrix Algebra, 2011a

3.1 The Matrix Transpose
symmetric.
(A − A T ) T = A T − (A T ) T transpose subtracon rule
= A T − A
= −(A − A T )
So we took the transpose of A − A T and we got −(A − A T ); this is the definion of
being skew symmetric.
We’ll take what we learned from Example 66 and put it in a box. (We’ve already
proved most of this is true; the rest we leave to solve in the Exercises.)
.
Ṫheorem 12
Symmetric and Skew Symmetric Matrices
.
1. Given any matrix A, the matrices AA T and A T A are symmetric.
2. Let A be a square matrix. The matrix A + A T is symmetric.
3. Let A be a square matrix. The matrix A − A T is skew
symmetric.
Why do we care about the transpose of a matrix? Why do we care about symmetric
matrices?
There are two answers that each answer both of these quesons. First, we are
interested in the tranpose of a matrix and symmetric matrices because they are interesng.
9 One parcularly interesng thing about symmetric and skew symmetric
matrices is this: consider the sum of (A + A T ) and (A − A T ):
(A + A T ) + (A − A T ) = 2A.
This gives us an idea: if we were to mulply both sides of this equaon by 1 2
, then the
right hand side would just be A. This means that
A = 1 2 (A + AT ) + 1
} {{ }
2 (A − AT ) .
} {{ }
symmetric skew symmetric
That is, any matrix A can be wrien as the sum of a symmetric and skew symmetric
matrix. That’s interesng.
The second reason we care about them is that they are very useful and important in
various areas of mathemacs. The transpose of a matrix turns out to be an important
9 Or: “neat,” “cool,” “bad,” “wicked,” “phat,” “fo-shizzle.”
129