# Homework Help: Matrices Questions

1. Sep 26, 2011

### Ted123

Is any symmetric matrix diagonalisable with an orthogonal change of basis?

Does the minimal polynomial of any real matrix split into distinct linear factors?

Is a real inner product an example of a bilinear form?

Could 2 complex matrices which are similar have different Jordan normal forms?

2. Sep 26, 2011

### micromass

So, what did you try already??

3. Sep 26, 2011

### Ted123

I just want to know 'yes' or 'no' out of interest

4. Sep 26, 2011

### micromass

For the first one, let A be symmetric. Let u be an eigenvector with eigenvalue $\lambda$ and let w be an eigenvector with eigenvalue $\mu$. Can you calculate

$$<Au,w>$$

in several ways?

5. Sep 26, 2011

### Ted123

I think YES, NO, YES, NO respectively. Is that correct?

6. Sep 26, 2011

### micromass

Why do you think that?

7. Sep 26, 2011

### Ted123

I think the 4th question is worded slightly ambiguously; it's probably safe to assume that it means "different up to reordering of the Jordan blocks" (in which case you're correct). But if it doesn't, then $$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{pmatrix}$$ and $$\begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}$$ would be an example of similar matrices with different JNFs.

I know the first 3 are right - yes, no, yes but for the 4th one, do you think that the answer to the last one is no?

8. Sep 26, 2011

### micromass

If you mean "different up to reordering of the Jordan blocks" then you are correct. Similar matrices have the same Jordan canonical form then.

This can be seen by putting $A=S^{-1}BS$. If C is the Jordan basis, then

$$CAC^{-1}$$

is the Jordan canonical form of A. Now, can you find easily the Jordan canonical form of B (just plug the expressions into each other).