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Matrix Transform

  1. Aug 27, 2010 #1
    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. jcsd
  3. Aug 27, 2010 #2

    marcusl

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    Gold Member

    Sounds like a homework problem. Please show your attempt at a solution.
     
  4. Aug 28, 2010 #3
    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.
     
  5. Aug 28, 2010 #4
    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
  6. Aug 28, 2010 #5
    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?
     
  7. Aug 28, 2010 #6

    Mark44

    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.
     
  8. Aug 28, 2010 #7
    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?
     
  9. Aug 28, 2010 #8
    You are right, I was wrong: I edited my post and now it should be correct. Are you familiar with Jordan decompositions of matrices?
     
  10. Aug 28, 2010 #9

    HallsofIvy

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    Staff Emeritus
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    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.
     
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