Problem $4$

Define a function norm$\_$approx$\_$gen $($ M$,$x$\_0,$k $)$ that approximates the

dominant eigenvector and eigenvalue of any square matrix. Your function

should take as input a square matrix m of any size, as well as an initial guess

x$00$ and integer k$,$ and use the power iteration method described above to

approximate the dominant eigenvector and eigenvalue of m up to the k th

iterate.

Your function should normalize at each step by dividing the eigenvector

approximation by the largest absolute value of its entries. You function should

return the approximation x$\_($k$)$ of the eigenvector obtained in this way, as well as

the eigenvalue approximation obtained by dividing the first entries of w$\_($k$)$ and

x$\_($k$-1).$

Quick Note

The command np$.$abs $($a$)$ returns a vector whose entries are the

absolute value of the entries in the NumPy vector a$.$ The command

np$.$ max $($a$)$ returns the maximum element of the NumPy vector a$.$ You

should also check this against simple examples for which you can

compute eigenvalues by hand, perhaps a $3$xx$3$ matrix from the

homework. Keep in mind that power iteration will only work for matrices

that have a dominant eigenvalue.

Test Cases

You can test that your code is correct by evaluating these commands and

comparing outputs:

norm$\_$approx$\_$gen$($np$.$array$([[2,4,6],[4,8,0],$

$[1,2,9]]),$ np$.$array$([1,5,-1]),10)$

$($array$([0\mathrm{.}98994349,1.,0\mathrm{.}98491523]),12\mathrm{.}01744017467949)$