Matrix Transform: Meaning & Space Covered

[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

 

[marcusl]

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

 

[Leo321]

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

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.

 

[Petr Mugver]

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?

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.

What space is covered by possible values of D for a given A?

This "space" is not a vectorial space. It is the set composed of all matrices that have the same Jordan decomposition of A.

What is the minimal set of matrices A, parametrized by as few as possible parameters, which covers all possible matrices D?

This set is composed by a representative matrix for each possible Jordan decomposition of a n x n matrix.

 

[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?

 

[Mark44]

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

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.

I multiply this by C^-1 from the left and C from the right and get:
A=I
What went wrong?

 

[Leo321]

It is the set composed of all matrices that have the same rank of A.

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.

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?

 

[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?

 

[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 R^m, 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.