Showing Symmetry in Real Invertible Matrices and Non-Invertible Cases

In summary, the conversation discusses how to show that the given product of vectors x and y, <x,y>=y'A'Ax, defines an inner product in R^n when A is a real invertible n * n matrix. The participants also mention the importance of symmetry, positive definiteness, and bilinearity in proving that this product is an inner product. They also mention the use of scalars, vectors, and matrices in their discussion.
  • #1
Benny
584
0
Hi can someone please help me get started on the following question?

Q. Let A be a real invertible n * n matrix. Show that [tex]\left\langle {\mathop x\limits^ \to ,\mathop y\limits^ \to } \right\rangle \equiv \mathop y\limits^ \to A^T A\mathop x\limits^ \to = \left( {A\mathop y\limits^ \to } \right)^T \left( {A\mathop x\limits^ \to } \right)[/tex] 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 a matrix M, obtained by intechanging the rows and columns of M).

The first step would be to show that the inner product is symmetric I would say. I think I should get to [tex]... = \mathop x\limits^ \to A^T A\mathop y\limits^ \to [/tex] but I don't know how to do get to it. Can someone suggest a method to use? I'm not sure if I need to explicity write down a matrix in this question.
 
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  • #2
(Ay)^t * (Ax) can be interpreted as the "usual" inner product of the vectors Ay and Ax...
 
  • #3
Let's drop the silly arrows from vectors, eh? vectors are lower case, matrices are upper case, we'll use ascii pseudo tex so that ' means transpose.

So, we want to show that <x,y>=y'A'Ax is a real inner product.

Firstly since <x,y> is a real number, ie a vector in 1-d then it is symmtric, ie <y,x>=<x,y>. Linearity is even easier since we're just multipliying matrices.
 
  • #4
You want to show that your given product is symmetric, positive definite, and multilinear. To show it's symmetric:

<x, y> = y'A'Ax
<y, x> = x'A'Ay = (x'A'Ay)' = y'A'Ax as required

The above basically says that a real number is like a 1x1 matrix, which is of course a symmetric matrix. I guess this is what you meant matt? To show it's positive definite

<x, x> = x'A'Ax = (Ax)'(Ax)

Ax is just a vector, and (Ax)'(Ax) is just the sum of the squares of the entries of Ax, so clearly <x, x> > 0 with equality iff x = 0. You can prove linearity on your own. Note you only have to prove linearity in one "component" and the fact that it is symmetric guarantees bilinearity. So just prove:

<ax + y, z> = a<x, z> + <y, z>
 
  • #5
the "unofficial convention" for scalars is to use letters like r,s,t (preferably greek letters like lambda but i can't do that in plain html) for them, reserving u,v,w,x,y,z for vectors and A,B for matrices.

and yes, that was what i meant, AKG about 1x1 matrices and real numbers being the same thing.
 
  • #6
Thanks for the help guys. When I write vectors I normally use a tilde, I just used arrows because I didn't know how to put a little squiggle underneath the vectors.
 

1. What is an invertible matrix?

An invertible matrix is a square matrix that can be reversed or "inverted" by multiplying it with another matrix. This results in the identity matrix, where every element on the main diagonal is 1 and all other elements are 0.

2. How do you show symmetry in real invertible matrices?

To show symmetry in real invertible matrices, you must first prove that the matrix is square and that its determinant is not equal to 0. Then, you must show that the matrix is equal to its transpose, which means that the elements on either side of the main diagonal are equal.

3. What is a non-invertible matrix?

A non-invertible matrix is a square matrix that cannot be reversed or "inverted" by multiplying it with another matrix. This is because the matrix does not have a determinant or its determinant is equal to 0.

4. How do you show symmetry in non-invertible cases?

In non-invertible cases, symmetry can still be shown by proving that the matrix is equal to its transpose. However, since the matrix is non-invertible, it will not result in the identity matrix.

5. What is the significance of showing symmetry in matrices?

Showing symmetry in matrices is important because it indicates that the matrix is well-behaved and has certain properties, such as being diagonalizable. It also helps in solving systems of equations and understanding the behavior of the matrix in various operations.

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