Question

I got a question:

If the eigenvallues and eigenvectors of a matrix A are Lambda_i, i = 1, 2 ,..., n and u_i, i = 1, 2, ... , n, respectively, what are the eigenvalues and eigenvectors of A^m where m is a positive integer.

I guess it is basically asking what happens to the eigenvalues and vectors of a matrix when you raise the matrix to the power m.

I think for the eigenvalues it raises them also to the power m but i'm unsure what happens to the vectors....
i always thought though that the eigenvectors/values would remain the same as long as the functions remained the same.

Answer 1

You are correct. The eigenvectors stay the same. The eigenvalues are raised to the power m. Raising A to the power M should not change the space that A is in, but it does change how "fast" solutions converge to their long-term solutions.

Answer 2

thanks. i was trying to verify this with mathematica but when i tell it to evaluate the cell it does something stupid. just returns the function with the stored matrix in it's variable. grrrr. oh well thanks for explaining.

Answer 3

It doesnt actually return the value.

Answer 4

In[24]:= B = MatrixPower[A,2]
Out[24]= MatrixPower[{{0.85787, 0.18619, 0.785592, 0.954878, 0.616075}, {0.107179, 0.0699923, 0.500129, 0.58611, 0.833044}, {0.733113, 0.757394, 0.0202323, 0.604503, 0.700401}, {0.7707, 0.413424, 0.322793, 0.523565, 0.169472}, {0.948852, 0.254859, 0.626333, 0.335197, 0.985757}},2]

Answer 5

where that mess in the output was A have you used mathematica @ybarrap ?

Answer 6

Try this, I've raised A to power 2 and see effect on eignenvectors and values:
http://www.wolframalpha.com/input/?i=eignvalues {{1%2C2%2C3}%2C{4%2C5%2C6}%2C{7%2C8%2C9}}^2

Answer 7

In case you can click on the url above, try this: http://tinyurl.com/dmiladin-1