Let me do this task from the previous post.
First you need to find eigenvalues.
##det
\begin{bmatrix}
2-\lambda&1\\
4&-1-\lambda
\end{bmatrix}
=0##
##(2-\lambda)(-1-\lambda)-1 \cdot 4=0##
##(2-\lambda)(-1-\lambda)- 4=0##
##-2-2\lambda+\lambda+\lambda^2-4=0##
##\lambda^2-\lambda-6=0##
I need to find solutions ##\lambda_1## and ##\lambda_2##. There are many ways to find them.
[1] Standard formula (copy/paste from
Forum > Help > LaTeX)
##\lambda_{1,2}= \frac {-b \pm \sqrt{b^2 -4ac}} {2a}##
[2] You can use Vieta's formulas. What are those two numbers?
When you add them together, you will get a minus coefficient in front of the lambda ##\lambda^2+(-1)\lambda-6=0##
##\lambda_1+\lambda_2=-(-1)=1##
When you multiply them, you will get the last number from the left side of the equation ##\lambda^2-1\lambda-6=0##
##\lambda_1 \cdot \lambda_2=-6##
Can you guess what those two numbers ( ##\lambda_1## and ##\lambda_2##) are?
[3] Complete to a "complete square". Add 1/4 to both sides of the equation
##\lambda^2-\lambda+\frac{1}{4}-6=0+\frac{1}{4}##
##\lambda^2-\lambda+\frac{1}{4}=6+\frac{1}{4}## ... left side is a "comlete square"
##(\lambda-\frac{1}{2})^2=\frac{24}{4}+\frac{1}{4}##
##(\lambda-\frac{1}{2})^2=\frac{25}{4}##
##\lambda-\frac{1}{2}=\pm \sqrt{\frac{25}{4}}##
##\lambda-\frac{1}{2}=\pm \frac{5}{2}##
##\lambda-\frac{1}{2}=\pm 2\frac{1}{2}##... two and a half
##\lambda_1=+ 2\frac{1}{2}+\frac{1}{2}=3##
##\lambda_2=- 2\frac{1}{2}+\frac{1}{2}=-2##
These are two eigenvalues.
Now I will find eigenvectors
[1] For eigenvalue ##\lambda_1=3## I get
##\begin{bmatrix}
2-3&1\\
4&-1-3
\end{bmatrix}
\begin{bmatrix}
x\\
y
\end{bmatrix}
=
\begin{bmatrix}
0\\
0
\end{bmatrix}
##
##\begin{bmatrix}
-1&1\\
4&-4
\end{bmatrix}
\begin{bmatrix}
x\\
y
\end{bmatrix}
=
\begin{bmatrix}
0\\
0
\end{bmatrix}
##
##-x+y=0 ##
##4x-4y=0##
These two equations are equivalent (have the same set of solutions). If you multiply the first by -4 you will get the second. There is no need to solve them both. It is enough to solve only one.
##y=x ##
The solution set is a straight line with infinitely many eigenvectors for the same eigenvalue ##\lambda_1=3##. I need to choose one.
For x=42 I get eigenvector ##v_1=(42,42)##
For x=1 I get eigenvector ##v_1=(1,1)##
...
[2] For eigenvalue ##\lambda_2=-2## I get
##\begin{bmatrix}
2-(-2)&1\\
4&-1-(-2)
\end{bmatrix}
\begin{bmatrix}
x\\
y
\end{bmatrix}
=
\begin{bmatrix}
0\\
0
\end{bmatrix}
##
##\begin{bmatrix}
4&1\\
4&1
\end{bmatrix}
\begin{bmatrix}
x\\
y
\end{bmatrix}
=
\begin{bmatrix}
0\\
0
\end{bmatrix}
##
##4x+y=0 ##
##4x+y=0##
These two equations are equivalent (have the same set of solutions) because they are the same .:-)
##y=-4x ##
The solution set is a straight line with infinitely many eigenvectors. I need to choose one.
For x=1 I get eigenvector ##v_2=(1,-4)##
That is it.