Solution to Problem15
Let A be a symmetric matrix.
- Show that A^2 is symmetric.
- Show that 2A^2 - 3A + I is symmetric.
Recall the definition of a symmetric matrix.
A matrix is symmetric if it equals its transpose.
We also need the following simple properties,
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:
- transpose(AB) = transpose(B) transpose(A)
- transpose( A + t B) = transpose(A) + t transpose(B)
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
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Carlos Rodriguez <email@example.com>
Last modified: Wed Feb 9 16:14:25 EST 2000