- #1

CGandC

- 326

- 34

**holds:**

__theorem__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 } ##

And according to Wolfram:

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

**above.**

__theorem__**:**

__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 )