# Inner product spaces of matrices (linear algebra)

• Luongo
So forget matrices and just do it as a calculation with vectors in R^4.In summary, the conversation discusses the understanding of the inner product in Rn and the vector space of C[a,b] as the integral operator. The question raised is how to obtain or prove the inner product space of two 2x2 matrices. The formula for the inner product is given as ae+2bf+3cg+4hd and the conversation suggests working out the axioms to prove that the subspace spanned by these two matrices is indeed an inner product. It is recommended to view the inner product as a formal manipulation and put aside intuition about parallel vectors and Euclidean spaces. The conversation also mentions using Lay's linear algebra book for reference

#### Luongo

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I understand the concepts of the inner product in Rn as well as the vector space of C[a,b] as the integral operator, however i don't understand how to obtain or prove the inner product space of two 2x2 matrices?
Example: consider two matrices u,v which are row 1 [a b] row 2 [c d]

and row 1[e f],row 2[g h]

where u is the 1st matrix and v is the second
for example if the inner product of <u,v>= ae+2bf+3cg+4hd
How do i satisfy the 4 axioms that prove the subspace spanned by these two matrices is infact an inner product?

Thanks.
P.S. i use lay's linear algebra book and there is NOTHING on matrices of inner products, only integral operators, R^n but not matrices and we weren't taught in class how...?

Since you are explicitly given the formula for the inner product, you can just work out the axioms.
You know that 2 x 2 matrices can be added, so calculate <u, v + w> where u and v are as given an w is [[m n] [p q]] and show that it is equal to <u, v> + <u, w>, using what you know about matrices.
Calculate <u, u> and show that it is non-negative, and zero iff a = b = c = d = 0.
And so on.

In this type of calculations, I think it is best to put aside your intuition about parallel vectors and Euclidean spaces, and simply view the inner product as just a formal manipulation of which you need to check that it has certain properties.

Luongo said:
How do i satisfy the 4 axioms that prove the subspace spanned by these two matrices is infact an inner product?

I don't see what that would accomplish: an inner product isn't a subspace.

P.S. i use lay's linear algebra book and there is NOTHING on matrices of inner products, only integral operators, R^n but not matrices and we weren't taught in class how...?

Since the 2x2 matrices are just a 4-dimensional vector space over R, then the book has taught you about inner products on them.

what does it mean when <u,v>=ae+2bf+3cg+4dh, how do you show the first 2 axioms,

The first two axioms of what? Axioms are not canonically ordered; I will not know exactly what is in your notes/book.

i don't understand the formula for the inner product it is not a matrix multiplication, it is just multiplying positions of the two matrices together, that is not how you multiply matrices.

Forget they're matrices. It has nothing to do with u and v being matrices other than that they live in a vector space.

For proving the first axiom, i just multiplied respective positions from each matrix ie: i got ea+fb+gc+hd. where do i get the scalar multiples of "2" "3" and "4" as given in the formula <u,v>=ae+2bf+3gc+4dh?

You just have to verify that something given to you satisfies the definition of an inner product. There is nothing deep about the formula you were given.

Forget matrices. Just think of u and v as vectors in R^4 by writing the 2 columns as a single column vector:

$$u=(a,b,c,d)^t$$

with the t for transpose, as it's easier to type row matrices. Show that

u.v=ae+2bf+3gc+4dh

is an inner product. It clearly is as it is just u^tAv where A is diag(1,2,3,4).