2 4 points a Prove that A B TraceAB defines an inner produ

2. (4 points) a. Prove that A * B = Trace(AB) defines an inner product on the space S^n of symmetric n x n matrices. b. Find an orthonormal basis for S^n in this inner product.

Solution

As per the Theorem Trace of AB , ie tr(AB) = tr(BA), when matrix follow the same number of row and column.

==> Let us say Matrix A is n xn then Matrix B also to be n X n.

b.

Let X be a vector space with inner product ?

Any mutually orthogonal non zero vectors in X is linearly independent.

Sⁿ with the dot product is an inner product spcae and the standard basis E = { e1,e2,