# Canonical form of matrices

1. Feb 23, 2008

### Physicsissuef

1. The problem statement, all variables and given/known data
Matrix:
$$\left| \begin{array}{ccc} \-1 & -2 & 5 \\ 6 & 3 & -4 \\ -3 & 3 & -11 \end{array} \right|\]$$
2. Relevant equations

3. The attempt at a solution

How will this matrix transfered into canonical form? What is actually canonical form?

Last edited: Feb 23, 2008
2. Feb 24, 2008

### HallsofIvy

Good question. How does your textbook define "canonical form"? Look it up in the index.

I ask for two reasons. First, you need to learn to look things up for yourself. Second, I'm not sure what you mean by "canonical" form. I know "Jordan canonical form" (also called "Jordan Normal form"), "rational canonical form", and "Frobenius canonical form". It's perfectly correct to use "canonical form" as long as you are talking about just one of those but I don't know which.

Last edited by a moderator: Feb 24, 2008
3. Feb 24, 2008

### Physicsissuef

In my book, says, turn that matrix with row transformations.
For example.
$$R_2\rightarrow 3*R_1+R_3$$
So I'll get:
$$\left| \begin{array}{ccc} \-1 & -2 & 5 \\ 6 & 0 & 0 \\ -3 & 3 & -11 \end{array} \right|\]$$

4. Feb 24, 2008

### HallsofIvy

Turn it into what? Triangular form? Row Echelon?

5. Feb 24, 2008

### Physicsissuef

Turn into canonical scale matrix. Like
$$\left| \begin{array}{ccc} \ 1 & -2 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array} \right|\]$$

6. Feb 24, 2008

### Physicsissuef

Do u know some other method of turning?

7. Feb 24, 2008

### HallsofIvy

I know how to do many different things by "row operations". I was trying to get you to tell what kind of "canonical" matrix you were talking about! It appears that you mean what I would call an "upper triangular matrix". Unfortunately, an example is not a definition (I've lost track of how many times I have told students that). In particular, you example has two 0s on the diagonal which is not, in general, possible. An "upper triangular matrix is a matrix that has only 0s below the main diagonal, but can have anything on or above it. But I don't see how
$$R_2\rightarrow 3*R_1+R_3$$
will accomplish that or what it is intended to accomplish. Could you please give me your definition of "canonical (scale) matrix" as I asked initially?

8. Feb 24, 2008

### Hurkyl

Staff Emeritus
While that is a row operation, it's not an elementary row operation, nor is it the product of such operations.

9. Feb 24, 2008

### HallsofIvy

Oh, you're right. I didn't even notice the change in index.

10. Feb 25, 2008

### Physicsissuef

In scale matrices, there are zeros like scales, it is not upper triangular matrix. So I can create scale with minimum 0 zero in one row, and +1 in the others.