MAE $284$ Spring $2024$ Homework

Assignment #$6(100$ pts$)$

Eigenvalues

Consider the following A matrix

$A=\left[\begin{array}{cc}-10& 9\\ -6& 5\end{array}\right]$

a$)$ Calculate the eigenvalues by hand using the equation $[I-A]=0.$ Use the equation editor to make the calculations easy to read.

Make sure to$,($i$)$ display the characteristic equation in the form ${}^2+a+b=0$ where you fill in the appropriate a & b values and $($ii$)$ the eigenvalues

b$)$ Calculate the eigenvectors by hand in a text field using the equation:

where

$\{\begin{array}{c}\text{:}[A-{}_{i}I]V_i=0\end{array}$

$V_i=\left[\begin{array}{c}{v}_i^1\end{array}\right]$

$v_i^2$

The superscripts are just used to indicate there are two elements. Make sure the eigenvectors are displayed and clearly marked. Do not normalize at this step.

c$)$ Calculate the eigenvalues and eigenvectors using the eig function $($display the output$).$

d$)$ In print statements show that eigenvalues and eigenvalues from parts a$-$c are same. You will most likely need to do adjustments to the $"$by hand" eigenvectors to do so$.$ I suggest using a format like the one below

The eigenvectors from MATLAB are $x$;$x$ and $x$;$x$

The eigenvectors by hand are $x$;$x$ and $x$;$x$