# Finding eigenvectors and canonical form.

So, I had a polynomial given as follows: $$f=x_1x_2-x_2x_3$$

( The task is to transform the quadratic form into canonical form, but I have to do it by using main theorem, I cannot do it by simply finding eigenvalues and then do a diagonal matrix with the corresponding values.

I constructed a corresponding matrix:

$$\begin{pmatrix}0 & \frac{1}{2} & 0 \\ \frac{1}{2} & 0 & -\frac{1}{2} \\ 0 & -\frac{1}{2} & 0\end{pmatrix}$$

Also, how can I determine the rank of the form, if there are three different eigenvalues? And one of them is zero?

Then I found eigenvalues: $$\lambda_1=0$$ and $$\lambda_{2,3}=\pm\frac{1}{\sqrt{2}}$$

Then I kind of struggle with eigenvectors.

First, with the $$\lambda_1=0$$
$$\frac{1}{2}y_1-\frac{1}{2}y_3=0$$ Suppose that $$y_1=1$$ then $$y_3=1$$ So the eigenvalue for $$\lambda_1=0$$ is (1,0,1)

Then with the $$\lambda_2=+\frac{1}{\sqrt{2}}$$ the matrix looks like

$$\begin{pmatrix}-\frac{1}{\sqrt{2}} & \frac{1}{2} & 0 \\\frac{1}{2} &-\frac{1}{\sqrt{2}} & -\frac{1}{2} \\0 & -\frac{1}{2} & -\frac{1}{\sqrt{2}}\end{pmatrix}$$

So, supposing that $$y_1=\frac{1}{\sqrt{2}}$$, $$y_2=1$$ and $$y_3= \frac{1}{\sqrt{2}}$$.

So the eigenvector when $$\lambda_2=+\frac{1}{\sqrt{2}}$$ is $$(\frac{1}{\sqrt{2}}, 1, \frac{1}{\sqrt{2}}$$) After some source it seems as the values for this eigenvector are incorrect.

For the third eigenvalue $$\lambda_3=-\frac{1}{\sqrt{2}}$$ the matrix looks like:

$$\begin{pmatrix}\frac{1}{\sqrt{2}} & \frac{1}{2} & 0 \\\frac{1}{2} &\frac{1}{\sqrt{2}} & -\frac{1}{2} \\0 & -\frac{1}{2} & \frac{1}{\sqrt{2}}\end{pmatrix}$$

If we suppose that $$y_1=-\frac{1}{\sqrt{2}}$$ then $$y_2=1$$ and $$y_3=\frac{\sqrt{2}}{2}$$

So that the eigenvector for $$\lambda_3=-\frac{1}{\sqrt{2}}$$ equals to $$(-\frac{1}{\sqrt{2}},1,\frac{\sqrt{2}}{2})$$

I suppose that computed values are incorrect, if so, can you give me the correct way to find them? Secondly how from there can I construct the canonic form? ( My main task was to transform the given polynomial from quadratic form to canonical form, but it was to be done by using the theorem).

• It would be useful if you could give more context. What is the main theorem ? What exactly are you supposed to do ? What is the ground field ? – Captain Lama May 30 at 21:00
• Very similar question to the one you have asked four hours ago.... math.stackexchange.com/q/3245417 – Jean Marie May 30 at 21:06
• @CaptainLama 1) Have to find eigenvalues. 2)I need to write a canonical form for a given quadratic form in the orthonormal eigenvector basis of matrix A ( the one constructed from given quadratic form). 3) Quadratic form in this base is diagonal matrix. I think that either I haven't computed the eigenvectors correctly or they are not orthogonal in this case. – Ieva Brakmane May 30 at 22:13