Matrices, cauchy-schwarz and tringla inqualities?

[LorkFFC]

Right I've been given 2 2x2 matrices
A =
|1 1|
|0 1|

B =
|1 0|
|1 1|

And I have to find the triangle
http://images.planetmath.org:8080/cache/objects/1628/js/img2.png
and cauchy schwarz
http://images.planetmath.org:8080/cache/objects/1628/js/img3.png
inequalities with the innerproduct defined as <A,B>=trace(ABt) where Bt is B transpose.

Now I've got no idea how to do it (I really hate matrices) so can someone push me in the right direction? thanks in advance!

 

[lippyka]

The Cauchy-Schwarz Inequality states that for any two vectors, the inner product of the two vectors must be less than or equal to the product of the magnitude of the two vectors. In this case, the inner product is given as <A,B>=trace(ABt). Therefore, we can rewrite the Cauchy-Schwarz Inequality as trace(ABt) <= ||A||*||B||. Now, if we look at the matrices A and B, we can see that A has a magnitude of √2 and B has a magnitude of √2. Therefore, our inequality reduces to trace(ABt) <= 2. Similarly, for the triangle inequality, we have that ||A+B|| <= ||A|| + ||B||. Again, since A and B have magnitudes of √2, this reduces to trace((A+B)t) <= 4, where (A+B)t is the transpose of A+B.