Canonical Form of Matrices: Understanding and Transforming

[Physicsissuef]

Homework Statement

Matrix:
$$\left| \begin{array}{ccc}<br /> \-1 & -2 & 5 \\<br /> 6 & 3 & -4 \\<br /> -3 & 3 & -11 \end{array} \right|\]$$

Homework Equations

The Attempt at a Solution

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

 

[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.

 

[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}<br /> \-1 & -2 & 5 \\<br /> 6 & 0 & 0 \\<br /> -3 & 3 & -11 \end{array} \right|\]$$

 

[HallsofIvy]

Turn it into what? Triangular form? Row Echelon?

 

[Physicsissuef]

Turn it into what? Triangular form? Row Echelon?

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

 

[Physicsissuef]

Do u know some other method of turning?

 

[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?

 

[Hurkyl]

$$R_2\rightarrow 3*R_1+R_3$$

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

 

[HallsofIvy]

$$R_2\rightarrow 3*R_1+R_3$$

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

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

 

[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.