1. ## Inner Product Properties

Let V = Mnn be the real vector space of all n x n matrices. If A and B
are in V, we define (A,B) = Tr(B^TA) where Tr stands for trace and T
stands for transpose. Prove that this function is an inner product on V.

I don't understand how I would prove that the function is an inner
product.

According to my book, an inner product must satisfy the following
properties:

a) (u,u) > 0; (u,u) = 0 iff u = 0v.
b) (v,u) = (u,v) for any u,v in V.
c) (u+v,w) = (u,w) + (v,w) for any u,v,w in V.
d) (cu,v) = c(u,v) for any u,v in V and c is a real scalar.

I plugged in u = Tr(BTA).
I get how I can prove the first property.
(u,u) would be > 0 because it is a square. And if u = 0, then (u,u) = 0.
I don't get how I would prove the other properties. What would I plug
in v as? Should A and B be u and v, or is there another way? Thank you

2. Hello,

Let's rephrase the properties with the notations your text uses =)

a) (A,B) > 0; (A,A) = 0 iff A = 0B. << be careful, it's iff, not if
b) (B,A) = (A,B) for any A,B in V.
c) (A+B,C) = (A,C) + (B,C) for any A,B,C in V.
d) (mA,B) = m(A,B) for any A,B in V and m is a real scalar.

A, B and C are matrices.

For example $({\color{blue}A+B},{\color{red}C})=\text{Tr}({\col or{red}C}^T({\color{blue}A+B}))$

Use basic properties such as linearity of the trace to prove them
Try to do them, and even if you have a slight doubt, don't hesitate to post it here.

3. b) (v,u) = (u,v) for any u,v in V.
Strange... I've learned that $(v,u)$ must equals $(\overline{u,v})$, but I agree that when $v$ and $u$ takes their values in $\mathbb{R}$ it makes no difference.

4. Originally Posted by arbolis
Strange... I've learned that $(v,u)$ must equals $(\overline{u,v})$, but I agree that when $v$ and $u$ takes their values in $\mathbb{R}$ it makes no difference.
That is for a complex inner product space.
In a real inner product space that is okay.

5. That explained everything! Thanks so much!