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Thread: Formula for matrix raised to power n?

  1. #1

    Formula for matrix raised to power n?

    What is the general formula for raising a square 2x2 matrix to a power such as 10 or 20?

  2. #2
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    Re: Formula for matrix raised to power n

    Quote Originally Posted by JohnfromTampa
    What is the general formula for raising a square 2x2 matrix to a power such as 10 or 20?
    If fear that the answer that you expect does not exist.
    That is not to say that there are no general algorithms for performing the operations.
    But they are not simple is the way a general formula is.
    If A is a square matrix then [tex]A^4[/tex] is found this way:
    [tex]\begin{array}{l} A^4 = A^3 A \\ A^3 = A^2 A \\ A^2 = AA \\ \end{array}[/tex].
    “A professor is someone who talks in someone else’s sleep”
    W.H. Auden

  3. #3
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    Re: Formula for matrix raised to power n?

    If you're up to finding Eigenvalues and Eigenvectors, there is a very nice way to proceed.

    http://mathworld.wolfram.com/Eigenvector.html
    "Unique Answers Don't Care How You Find Them." - Many may have said it, but I hear it most from me.

  4. #4

    Re: Formula for matrix raised to power n?

    The solution is the product of three matrices: eigenvectors as the columns of a non-singular matrix T, diag the diagonal matrix of eigenvalues, and the inverse of T.

    A[sup:2lj8ud76]k[/sup:2lj8ud76]=Tdiag[sup:2lj8ud76]k[/sup:2lj8ud76]T[sup:2lj8ud76]-1[/sup:2lj8ud76]

    diag[sup:2lj8ud76]k[/sup:2lj8ud76]=[[tex]a[/tex][sup:2lj8ud76]k[/sup:2lj8ud76], 0; 0, [tex]d[/tex][sup:2lj8ud76]k[/sup:2lj8ud76]]

    For example, a matrix A=[2,1;0,3] has eigenvalues [tex]a[/tex], [tex]d[/tex] of 2,3 and eigenvectors of [1;0] and [1;1]. Then T=[1,1;0,1]

    A[sup:2lj8ud76]k[/sup:2lj8ud76]=[1,1;0,1][2,0;0,3][sup:2lj8ud76]k[/sup:2lj8ud76][1,1;0,1][sup:2lj8ud76]-1[/sup:2lj8ud76]

    A[sup:2lj8ud76]k[/sup:2lj8ud76]=[1,1;0,1][2[sup:2lj8ud76]k[/sup:2lj8ud76],0;0,3[sup:2lj8ud76]k[/sup:2lj8ud76]][1,-1;0,1], and

    A[sup:2lj8ud76]k[/sup:2lj8ud76]=[2[sup:2lj8ud76]k[/sup:2lj8ud76],3[sup:2lj8ud76]k[/sup:2lj8ud76]-2[sup:2lj8ud76]k[/sup:2lj8ud76];0,3[sup:2lj8ud76]k[/sup:2lj8ud76]]

  5. #5
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    Re: Formula for matrix raised to power n?

    Well, let's not get all arrogant and decide that we have THE WAY to solve the problem. Here's another...

    Using a little Cayley-Hamilton Theorem (A square matrix satisfies its own Characteristic Equation)...
    Throwing in a little Division Algorithm for Polynomials (or Remainder Theorem, if you like)...

    We get this remarkable result:

    If A is a 2x2 matrix, and If [tex]\lambda_{1}[/tex] and [tex]\lambda_{2}[/tex] are the distinct Eigenvalues, then we have

    [tex]A^{n}\;=\;\frac{\lambda_{2}(\lambda_{1})^{n}-\lambda_{1}(\lambda_{2})^{n}}{\lambda_{2}-\lambda_{1}}I_{2}\;+\;\frac{(\lambda_{2})^{n}-(\lambda_{1})^{n}}{\lambda_{2}-\lambda_{1}}A[/tex]

    It's a little different if the Eigenvalues are equal. Obviously, it takes more than that for 3x3 or larger.

    Never think you have the ONLY way unless you have PROVEN it so.
    "Unique Answers Don't Care How You Find Them." - Many may have said it, but I hear it most from me.

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    Re: Formula for matrix raised to power n?

    Quote Originally Posted by tkhunny
    Well, let's not get all arrogant and decide that we have THE WAY to solve the problem. Here's another...

    Using a little Cayley-Hamilton Theorem (A square matrix satisfies its own Characteristic Equation)...
    Throwing in a little Division Algorithm for Polynomials (or Remainder Theorem, if you like)...

    We get this remarkable result:

    If A is a 2x2 matrix, with elements a, b, c, and d, and

    If [tex]\lambda_{1}[/tex] and [tex]\lambda_{2}[/tex] are the distinct Eigenvalues, then we have

    [tex]A^{n}\;=\;\frac{\lambda_{2}(\lambda_{1})^{n}-\lambda_{1}(\lambda_{2})^{n}}{\lambda_{2}-\lambda_{1}}I_{2}\;+\;\frac{(\lambda_{2})^{n}-(\lambda_{1})^{n}}{\lambda_{2}-\lambda_{1}}A[/tex]

    It's a little different if the Eigenvalues are equal. Obviously, it takes more than that for 3x3 or larger.

    Never think you have the ONLY way unless you have PROVEN it so.
    THAT is why I want to be a math major - can't wait for linear algebra :P

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    Re: Formula for matrix raised to power n?

    Haha, I was at the bookstore and randomly opened up a book on tensor calculus. You'd think I'd be scared but it looks so interesting o.O. Although I'd love to major in mathematics, it wouldn't be ideal for me ... *sigh*

  8. #8

    Re: Formula for matrix raised to power n?

    No arrogance here. I'm just a student trying to learn. I thought I'd post a solution rather than leaving it hanging open.

  9. #9
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    Re: Formula for matrix raised to power n?

    To be fair, I was just picking on your choice of words, not your personal attitude. Good work.
    "Unique Answers Don't Care How You Find Them." - Many may have said it, but I hear it most from me.

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