Write a function

function T $=$ Spectral$\_$Decomposition$\_$Lambda$($ T $)$

That returns

$\backslash$Lambda

such that

T

$=$

Q

$\backslash$Lambda

Q

T

is the Spectral Decomposition of

T

$.$

The input matrix

T

is a tridiagonal matrix where only the lower triangular part of the matrix is stored in the diagonal and first subdiagonal of array T$.$ The diagonal matrix

$\backslash$Lambda

is returned in T $.$ The upper triangular part of the array should not change values. You are encouraged to call the function Francis$\_$Step from the function Spectral$\_$Decomposition$\_$Lambda. Obviously, you need to incorporate deflation in your implementation. How to handle the final

$2$

$\backslash$times

$2$

matrix is an interesting question... $($You may use the matlab function eig for this.$)$ Homework $10\mathrm{.}5\mathrm{.}1\mathrm{.}1.$ You may want to do a new "git pull" to update directory

In Assignments$/$Week$10/$matlab you will find the files

Givens$\_$rotation.m : A function that computes a Givens' rotation from a

$21$ vector x $.$

Francis$\_$Step.m : A function that performs a Francis Implicit QR Step with

a tridiagonal matrix $T($stored as the diagonal and subdiagonal of T $).$

Test$\_$Francis$\_$Step.m : A very rudimentary script that performs a few calls

to the function Francis$\_$Step. Notice that our criteria for the routine

being correct is that the matrix retains the correct eigenvalues.

With this,

Investigate the convergence of the $(m,m-1)$ element of matrix

Write a function

function T $=$ Spectral$\_$Decomposition$\_$Lambda$($ T $)$

That returns $$ such that $T=Q{Q}^T$ is the Spectral Decomposition of $T.$

The input matrix $T$ is a tridiagonal matrix where only the lower triangular

part of the matrix is stored in the diagonal and first subdiagonal of array

$T.$ The diagonal matrix $$ is returned in $.$ The upper triangular part

of the array should not change values. You are encouraged to call the

function Francis$\_$Step from the function

Spectral$\_$Decomposition$\_$Lambda $.$ Obviously, you need to incorporate

deflation in your implementation. How to handle the final $22$ matrix is

an interesting question... $($You may use the matlab function

for

this.$)$

PLEASE WRITE THE MATLAB CODE EXPLICITLY