Question 15127
 A : nxn square matrix over R. (missing)
 
A^T denotes the transpose of a matrix A.
Show that = Tr(B^T,A) defines an inner product on {{{R^(n x n) }}}
Let U be the set of symmetric 2 x 2 matrices with real entries. U is subspace of R (2x2). Find an orthonormal basis of U with respect to the above inner product.

 Let A = (aij), B = (bij) be nxn sq. matrices over R. ({{{SIGMA}}} means summation)
 Define (A,B) = {{{Tr(B^(T) A) }}}= {{{ SIGMA }}}{{{SIGMA}}} {{{ b[ki] a[ki] }}}(k=1,2..,n)(i=1,2..,n)
 
 You have to check the definition of inner product (as symmetric
 , linear, positive definite, all very easy)
 
 Note the dim of the vector space {{{R^(2x2)}}} = 4.
 And dim U = 3 (why?)
 Let A=
 (1 0)
 (0 0)
 B=
 (0 0)
 (0 1)
 C =
 (0 {{{1/sqrt(2) }}} )
 ({{{1/sqrt(2)}}} 0)
 then {A,B,C} forms an o.n. basis of U.

 You should test (A,A)=(B,B)=(C,C)= 1.
  (A,B) = (B,C) =(C,A) = 0

 Kenny