Solution to Problem15

Problem15:

Let A be a symmetric matrix.

Show that A^2 is symmetric.
Show that 2A^2 - 3A + I is symmetric.

SOLUTION:

Recall the definition of a symmetric matrix. A matrix is symmetric if it equals its transpose. We also need the following simple properties,

transpose(AB) = transpose(B) transpose(A)
transpose( A + t B) = transpose(A) + t transpose(B)

To show that A^2 = A A is symmetric when A is symmetric we only need to use the first of the properties above. To show that 2A^2 - 3A + I is symmetric we use the second property and the facts that A, A^2, and I are symmetric. In painful detail:

> ;

                        2           T       2 T      T    T
                    (2 A  - 3 A + I)  = 2 (A )  - 3 A  + I

and since A, A^2 and I are symmetric,

> ;

                          2           T      2      
                      (2 A  - 3 A + I)  = 2 A  - 3 A  + I

Link to the commands in this file

Carlos Rodriguez <carlos@math.albany.edu>

Last modified: Wed Feb 9 16:14:25 EST 2000