Congruence for Symmetric and non-Symmetric Matrices for Quadratic Form

In summary, the following theorem holds: for a bilinear form/square form, the matrices ## A,B ## are congruent if and only if ## A,B ## represent the same bilinear/quadratic form.
  • #1
CGandC
326
34
I learned that for a bilinear form/square form the following theorem holds:
matrices ## A , B ## are congruent if and only if ## A,B ## represent the same bilinear/quadratic form.

Now, suppose I have the following quadratic form ## q(x,y) = x^2 + 3xy + y^2 ##. Then, the matrix representing this quadratic form can be ## B = \pmatrix{1 & 3 \\ 0 & 1 } ## and also ## A = \pmatrix{1 & \frac{3}{2} \\ \frac{3}{2} & 1 } ##.

I tried showing that ## A, B ## are congruent, meaning that there exists an invertible matrix ## M ## such that ## B = M^T \cdot A \cdot M ##, but such matrix ## M ## does not exist, here's the calculation for example:
## M = \pmatrix{ a & b \\ c & d } ##
1670931401506.png

And according to Wolfram:
1670931427316.png


Yet, ## A,B ## represent the same quadratic form, but they are not congruent, so this is a contradiction to the theorem above.

Question:
How is this possible? These matrices should be congruent since they do represent the same quadratic form.
( And yes, I know that for every matrix ## A ##, ## \frac{1}{2} ( A + A^T ) ## is symmetric, but this doesn't answer why the above congruence fails to hold )
 
Physics news on Phys.org
  • #2
##B## does not represent the quadratic form.
 
  • #3
martinbn said:
##B## does not represent the quadratic form.
Yes it does because: ## \pmatrix{ x & y } \pmatrix{1 & 3 \\ 0 & 1 } \pmatrix{ x \\ y } = \pmatrix{ x & y }\pmatrix{x + 3y \\ y } = x(x+3y) +y^2 = x^2 + 3xy + y^2 ##
 
  • #4
I think you have some terminology confusion.

Given a bilinear form on a vector space, if you pick a basis it has a unique symmetric representation.

Two matrices are congruent if they are representatives of the same bilinear form with different choices of basis.

Note the representatives are always symmetric.
 
  • #5
Office_Shredder said:
I think you have some terminology confusion.

Given a bilinear form on a vector space, if you pick a basis it has a unique symmetric representation.

Two matrices are congruent if they are representatives of the same bilinear form with different choices of basis.

Note the representatives are always symmetric.
You mean a quadratic form?, because bilinear form isn't necessarily symmetric
 
  • #6
CGandC said:
You mean a quadratic form?, because bilinear form isn't necessarily symmetric
Sorry, you are correct, quadratic form
 
  • #7
Ok but if
## \pmatrix{ x & y } B\pmatrix{ x \\ y } = \pmatrix{ x & y } \pmatrix{1 & 3 \\ 0 & 1 } \pmatrix{ x \\ y } = \pmatrix{ x & y }\pmatrix{x + 3y \\ y } = x(x+3y) +y^2 = x^2 + 3xy + y^2 ##
And
## \pmatrix{ x & y } A\pmatrix{ x \\ y } = \pmatrix{ x & y } \pmatrix{1 & \frac{3}{2} \\ \frac{3}{2} & 1 } \pmatrix{ x \\ y } = \pmatrix{ x & y }\pmatrix{x + \frac{3}{2}y \\ \frac{3}{2}x+y } = x(x+\frac{3}{2}y) +y(\frac{3}{2}x+y) ##
## = x^2 + \frac{3}{2}xy + \frac{3}{2}yx + y^2 = x^2 +3xy+y^2 ##

Then according to what you said ## A ## is a representative of the quadratic form since it is symmetric.
But what is ## B ## then? is it also defined as a representative of the quadratic form?
 
  • #8
Ok, I've asked on mathexchange and I was told that when dealing with symmetric bilinear-forms/quadratic-forms then when using the theorems related to these, one should assume he's looking at the symmetric representation of these forms, otherwise most of the theorems for symmetric bilinear-forms/quadratic-forms will fail to hold and it'll be nonsense.

For example, in the above example if one assumes ## B = M^T A M ## to be symmetric then ## B^T = M^T A^T M = M^T A M = B ##, but this is a contradiction since ## B^T \neq B ##, so ##A,B ## are not congruent.
So everything's clear now, thanks for the help!
 

Related to Congruence for Symmetric and non-Symmetric Matrices for Quadratic Form

1. What is the definition of congruence for symmetric and non-symmetric matrices for quadratic form?

Congruence for symmetric and non-symmetric matrices for quadratic form is a mathematical concept that describes the similarity between two matrices when they have the same quadratic form. This means that the matrices have the same eigenvalues and eigenvectors, but their entries may be different.

2. How do you determine if two matrices are congruent for quadratic form?

To determine if two matrices are congruent for quadratic form, you can use the congruence transformation. This involves multiplying one matrix by a non-singular matrix and its transpose, and then comparing the resulting matrix to the other original matrix. If they are equal, then the matrices are congruent for quadratic form.

3. Can a symmetric matrix be congruent to a non-symmetric matrix for quadratic form?

Yes, a symmetric matrix can be congruent to a non-symmetric matrix for quadratic form. This is because congruence for quadratic form only requires the matrices to have the same eigenvalues and eigenvectors, not the same entries.

4. What are the applications of congruence for symmetric and non-symmetric matrices for quadratic form?

Congruence for symmetric and non-symmetric matrices for quadratic form has various applications in mathematics, physics, and engineering. It is used in diagonalization of matrices, solving systems of linear equations, and in optimization problems.

5. Is congruence for symmetric and non-symmetric matrices for quadratic form the same as similarity?

No, congruence and similarity are not the same concepts. While congruence for quadratic form looks at the eigenvalues and eigenvectors of the matrices, similarity looks at the linear transformations that can be represented by the matrices. In other words, congruence is a property of the matrices themselves, while similarity is a property of the transformations they represent.

Similar threads

  • Linear and Abstract Algebra
Replies
4
Views
2K
  • Linear and Abstract Algebra
Replies
13
Views
1K
  • Linear and Abstract Algebra
Replies
6
Views
1K
  • Linear and Abstract Algebra
Replies
2
Views
756
  • Linear and Abstract Algebra
Replies
2
Views
974
  • Linear and Abstract Algebra
Replies
8
Views
1K
  • Linear and Abstract Algebra
Replies
16
Views
1K
  • Linear and Abstract Algebra
Replies
3
Views
944
  • Linear and Abstract Algebra
Replies
5
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
1K
Back
Top