Matrix Powers - General Expression for Mk to the Nth Power

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In summary, The problem is to find a general expression for Mk to the nth power in terms of k and n, where Mk is a matrix in the form (k+1 k-1) (k-1 k+1). The conversation suggests using a systematic approach of diagonalizing the matrix to find the general expression.
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masterprimus
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This problem first appeared on another thread under Statistics and probability. I found it when I got the same problem, which is as follows

Consider matrices in the form (k+1 k-1)
(k-1 k+1)

We will call this matrix Mk, find a general expression for Mk to the nth power in terms of k and n.


I tried several different matrices of this form, the general expression i came up with is attached in the thumbnail, but I still tried making it somewhat clear with latex.

M[tex]^{N}_{K}[/tex] = 2[tex]^{n-1}[/tex][tex]\left( [(k+1) +(k - 1)\sum^{n}_{x=1} k^{x}] [(k-1) +(k - 1)\sum^{n}_{x=1} k^{x}] \right)[/tex]
[tex]\left([(k - 1) +(k - 1)\sum^{n}_{x=1} k^{x}] [(k + 1) +(k - 1)\sum^{n}_{x=1} k^{x}] \right)[/tex]
 

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I don't think that's quite right, if that was your question. But there is a systematic way to do this. M can be diagonalized since it's symmetric. Find the matrix which diagonalizes it so M=S^(-1).D.S where D is diagonal. Then M^n=S^(-1).D^n.S.
 
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1. What is a matrix power?

A matrix power is the result of multiplying a matrix by itself a certain number of times. It is denoted by raising the matrix to a power, such as M^n where M is the original matrix and n is the number of times it is multiplied by itself.

2. What is the general expression for Mk to the Nth power?

The general expression for Mk to the Nth power is M^(k*N), where M is the matrix and k is the number of times it is multiplied by itself. This means that each element of the original matrix will be raised to the power of k*N.

3. How do you calculate a matrix power?

To calculate a matrix power, you need to first determine the dimensions of the matrix. Then, you multiply the matrix by itself the specified number of times, following the rules of matrix multiplication. You can also use a calculator or computer program to calculate the matrix power for larger matrices.

4. What is the purpose of calculating matrix powers?

Calculating matrix powers can help solve complex mathematical problems, such as systems of linear equations. It is also used in various fields of science and engineering, such as in computer graphics, signal processing, and quantum mechanics.

5. Are there any limitations to calculating matrix powers?

Yes, there are limitations to calculating matrix powers. The dimensions of the matrix must be compatible for multiplication (i.e. the number of columns in the first matrix must equal the number of rows in the second matrix). Additionally, not all matrices have a defined matrix power, such as singular or non-square matrices. It is important to check for these limitations before attempting to calculate a matrix power.

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