# Math Help - Another Inner Product Question.

1. ## Another Inner Product Question.

Let A be a real invertible $n$ x $n$ matrix. Show that

<x,y> ≡ $y^T$ $A^T$ $Ax$ = $(Ay)^T$ $(Ax)$

defines an inner product in ℝ^n, where x and y are column vectors in ℝ^n. What happens when A is not invertible? (Note : $M^T$ is the transpose of matrix M, obtained by interchaging the rows and colums of M).

2. Anyone at all :\?

3. Hi

What has to be shown : ( $x,\,y,\,\in\mathbb{R}^n$, $\alpha\in \mathbb{R}$)
• $\,=\,$ (use $^tA\cdot B=^t( ^tB\cdot A)$, and remember that two $1 \cdot 1$ matrices (=two real numbers) are equal iff their transposes are equal)
• $\,=\,+$
• $<\alpha x,y>\,=\,\,=\,\alpha$
• $\,\geq \, 0$ and $\,=\,0 \Leftrightarrow x=0$. To show both, use $\,=\,^t(Ax)Ax$ and the definition of the product of matrices which involves a sum : $(Ax)_{i}=\sum_{k=1}^n a_{i,k}x_k$

4. Hmm, I still have no idea how to approach this :\ Can you demonstrate how to do the first property and the last property for me to see? Perhaps I could understand from there. Thanks!

5. Let $x,\,y\in\mathbb{R}^n$.

First one :
• $\,=\,^ty ^tAAx\,=\, ^t(^t(Ax)Ay)$
• $\,=\,^tx ^tAAy\,=\, ^t(Ax)Ay$
But $$ is a real numbers (i.e. $1\cdot 1$ matrix) hence $^t\,=\,$ and we get what we want.

Last one :
Denote $Ax=(Ax_i)=\left(\begin{array}{c} Ax_1\\ \vdots \\Ax_n\end{array}\right)$, $x=\left(\begin{array}{c} x_1\\ \vdots \\x_n\end{array}\right)$ and $A=(a_{i,j})$.
• $\,=\,^t(Ax)Ax=\sum_{i=1}^n(Ax)_i^2\geq 0$
• If $x=0$, we check that $\,=\,0$ because $(Ax)_i=\sum_{k=1}^na_{i,k}x_k=0$
• If $\,=\,0\,=\,\sum_{i=1}^n(Ax)_i^2,$ hence, for all $i$ such that $1\leq i \leq n$, $(Ax)_i=0$. This exactly means that $x$ is in the nullity of $A$. As $A$ is invertible, $x=0$.

6. Hmm, sorry but I still don't quite understand

Do I have to put like,

x = [ x1 x2 and y = [ y1 y2
x3 x4] y3 y4]

and insert them into whatever you provided above and work from there?

This is such a tough question

7. Originally Posted by pearlyc
Do I have to put like,

x = [ x1 x2 and y = [ y1 y2
x3 x4] y3 y4]

and insert them into whatever you provided above and work from there?
No, there is no need to add this. (at least, I don't know why it would be interesting to introduce this)
This is such a tough question
Tell me what you don't understand and I'll try to explain it or to make the post #5 clearer.

8. Hmm, I don't know how did we progress from the question,

<x,y> ≡ =

to

Can you show me how to do the other two as well ? Properties 2 & 3.

9. Originally Posted by pearlyc
Hmm, I don't know how did we progress from the question,

<x,y> ≡ =

to

There may have a problem of notation : I write $^tA$ the transpose of $A$. Anyway :

$\,=\,^ty ^tAAx$ (that's the definition)

$\,=\,^t(Ay)Ax$ because $^t(BC)=^tC\cdot ^tB$ for any two matrices $B$ and $C$.

hence $\,=\,^t(^t(Ax)Ay)$ because $^t( ^t(Ax)Ay)= ^t(Ay) \cdot^t( ^t(Ax))\,=\, ^t(Ay)Ax$ (previous rule with $B=^t(Ax)$ and $C=Ay$)

Can you show me how to do the other two as well ? Properties 2 & 3.
No. These ones are really basic questions that you should be able to answer by yourself. The only things you need to know is that $A\cdot (B+C)= A\cdot B+A\cdot C$ and that $(\lambda B)\cdot C=B\cdot(\lambda C)=\lambda (B\cdot C)$ with $A,\,B,\,C\in\mathcal{M}_n(\mathbb{R})$ and $\lambda\in\mathbb{R}$. Try to do it and, if you want, post your answer, I'll tell you if it's correct or not.

10. Okay, I have a slightly clearer view of it now considering the confusion was about the notation. Hm, to prove Property 1 only require that two steps ?

I don't know how to explain how did it go from <x,y> to <y,x>.

I am quite doomed right now, this is my assignment question which dues tomorrow and I am pretty much still quite blur about it

11. Originally Posted by pearlyc
I don't know how to explain how did it go from <x,y> to <y,x>.
We got $\,=\, ^t(^t(Ax)Ay)$ and we also know that $$ is a real number so it can be seen as the one-one matrix $\begin{pmatrix}\end{pmatrix}$. This gives us $^t\,=
\,^t\begin{pmatrix}\end{pmatrix}
=\begin{pmatrix}\end{pmatrix}
=\,$
.
Hence $\,=\,^t\,=\, ^t\left( ^t( ^t(Ax)Ay)\right)= ^t(Ax)Ay=$.

I am quite doomed right now, this is my assignment question which dues tomorrow and I am pretty much still quite blur about it
Does your assignment only depend on one question ?

12. Unfortunately, it does depend on this question.

So stressed out! I still don't know how to do.

13. Originally Posted by pearlyc
Unfortunately, it does depend on this question.
I know, you said it. I wanted to know it it *only* depends on it. (as you said "*my* assignment question")

So stressed out! I still don't know how to do.
What don't you know how to do ? The two remaining properties ? Show me what you've tried and I'll help you.

14. Sigh. It is quite bad shape now.

What for ? $x$ is a vector that's to say a $n\cdot 1$ matrix. What would it change to say "x=a matrix" ?