# Matrix Transform

1. Aug 27, 2010

### Leo321

Hi,
Suppose I have an nxn matrix A. (If needed it can be assumed invertible). I can perform a transform on the matrix in the following way:
D=C*A*C^-1. C can be chosen to be any nxn invertible matrix.
Does this transform have any meaning, which can be easily understood or visualized?
What space is covered by possible values of D for a given A?
What is the minimal set of matrices A, parametrized by as few as possible parameters, which covers all possible matrices D?
2x2 case is of particular interest, but a general answer would certainly be useful.

Thanks

2. Aug 27, 2010

### marcusl

Sounds like a homework problem. Please show your attempt at a solution.

3. Aug 28, 2010

### Leo321

This is not a homework problem.
I noticed that I formulated it the way homework/exam questions are often formulated with multiple paragraphs, but that's just because I am trying to fully understand what is going on here.

4. Aug 28, 2010

### Petr Mugver

This is the usual similarity transformation of homomorphisms: you can see both A and D as the representations of the same linear application with respect to different bases. Let's say A is the representation with respect to a base B, and D is the representation with respect to a basis E. Then the matrix C's columns are the vectors of the base B represented in the base E.
This "space" is not a vectorial space. It is the set composed of all matrices that have the same Jordan decomposition of A.
This set is composed by a representative matrix for each possible Jordan decomposition of a n x n matrix.

Last edited: Aug 28, 2010
5. Aug 28, 2010

### Leo321

Thanks.
If A is invertible, I should be able to find C, which transforms it into the identity matrix, right?
C*A*C^-1=I
I multiply this by C^-1 from the left and C from the right and get:
A=I
What went wrong?

6. Aug 28, 2010

### Staff: Mentor

No, you don't necessarily get the identity matrix. What you get under certain conditions is a diagonal matrix, one whose entries off the main diagonal are zero.

7. Aug 28, 2010

### Leo321

Don't these two claims contradict? If I can transform A into any matrix of the same rank, then if A has maximal rank, shouldn't I be able to transform it into the identity matrix?

8. Aug 28, 2010

### Petr Mugver

You are right, I was wrong: I edited my post and now it should be correct. Are you familiar with Jordan decompositions of matrices?

9. Aug 28, 2010

### HallsofIvy

More general than matrices is the "linear transformation" from one vector space to another.

Any matrix can be thought of as a linear transformation specifically from the vector space Rn to Rm, Euclidean spaces.

In the other direction, a linear transfromation from finite dimensional vector space U to finite dimensional vector space V can be written as matrix by selecting particular ordered bases in U and V.

Two matrices, A and B, say, represent the same linear transformation, as written using different bases, if and only if they are "similar": B= CAC-1 for some invertible matrix C.