Matrix Vector

(c) Matrix Vector

Problem 1.

Solution:

Thus, all the axioms are satisfied.

Hence the proof.

Problem 2.

Use the Frobenius inner product to compute || A ||, || B ||, and <A, B> for

Solution:

Problem 3.

Let A and B be n x n matrices, and let c be a scalar. Prove that 

Solution :

Let A and B be n x n matrices and c be a scalar.

The conjugate transpose of n x n matrix A is n xn matrix A* such that 

Problem 4.

Solution :

By the definition of the standard inner product

Problem 5

Solution :

Problem 6.

Solution :

The inner product <A, A> is given by,

Here, <A, B> = 0 which is a contradiction to axiom (d) of inner product.

Therefore, <A, B> = tr (A + B) is not an inner product on M_2x2(R)