helloo!
find the eigenvalues and eigen vectors of
A= |1 2|
|5 4|
i dont think its long to work out, but ive just started this topic and cannot work it out
A non zero vector x is called an eigenvector of A if Ax is a scalar multiple of x.
$\displaystyle Ax=kx$
Where k is the eigenvalue.
They're tied together. Lambda is mostly used, but I will use k.
You know your eigenvalues, -1 and 6. They're the roots of your characteristic polynomial.
$\displaystyle \begin{bmatrix}1&2\\5&4\end{bmatrix}-\overbrace{(-1)}^{\text{eigenvalue}}\begin{bmatrix}1&0\\0&1\end {bmatrix}=\begin{bmatrix}2&2\\5&5\end{bmatrix}$
$\displaystyle \begin{bmatrix}1&2\\5&4\end{bmatrix}-6\begin{bmatrix}1&0\\0&1\end{bmatrix}=\begin{bmatr ix}-5&2\\5&-2\end{bmatrix}$
$\displaystyle rref\begin{bmatrix}2&2\\5&5\end{bmatrix}=\begin{bm atrix}1&1\\0&0\end{bmatrix}$
$\displaystyle rref\begin{bmatrix}-5&2\\5&-2\end{bmatrix}=\begin{bmatrix}1&\frac{-2}{5}\\0&0\end{bmatrix}$
In the first matrix, we have $\displaystyle x=-y$
Let y=t, then we have x=-t and y=t
In the second one, we have $\displaystyle x=\frac{2}{5}y$
Let y=t, then $\displaystyle x=\frac{2}{5}t$
Therefore, the eigenvectors are
$\displaystyle \begin{bmatrix}-1\\1\end{bmatrix}\cdot{t}$
and
$\displaystyle \begin{bmatrix}\frac{2}{5}\\1\end{bmatrix}\cdot{t}$
i understand how to get upto here, but i dont see how you would get the eigenvectors because the teacher didnt explain the steps. i thought that for the 2255 matrix, it would be 2X1+2X2=0 so x1=-X2 so in the eigenvector, the top coordinate would be -1 and the same principle works for the bottom so it would be -1 again so the eigenvector would be (-1/-1) but i think this is wrong...
Let's try this in a different way. We know the eigenvalues are -1 and 6. So we may write the equation for the -1 eigenvector as (from the eigenvalue equation $\displaystyle Ax = \lambda x$):
$\displaystyle \left [ \begin{array}{cc} 1 & 2 \\ 5 & 4 \end{array} \right ] \cdot \left [ \begin{array}{c} a \\ b \end{array} \right ] = (-1) \left [ \begin{array}{c} a \\ b \end{array} \right ] $
So we have the simultaneous equations
$\displaystyle a + 2b = -a$
and
$\displaystyle 5a + 4b = -b$
Solving this for a and b gives a = a and b = -a. So the eigenvector is
$\displaystyle \left [ \begin{array}{c} a \\ -a \end{array} \right ] = a \left [ \begin{array}{c} 1 \\ -1 \end{array} \right ]$
(This is just the negative of galactus' form for the eigenvector. Since the variable a is unspecified we may easily set t = -a and get my form from his.)
We may get an unambiguous eigenvector if we have some condition on the eigenvector, such as orthonormality or something. Since this wasn't mentioned I will leave the eigenvector like this. The 6 eigenvalue is similar:
$\displaystyle \left [ \begin{array}{cc} 1 & 2 \\ 5 & 4 \end{array} \right ] \cdot \left [ \begin{array}{c} a \\ b \end{array} \right ] = (6) \left [ \begin{array}{c} a \\ b \end{array} \right ] $
I'll leave it to you to work out the rest.
-Dan
You are getting an eigenvector of
$\displaystyle a \left [ \begin{array}{c} 1 \\ \frac{5}{2} \end{array} \right ] $
Again, this is simply a modification of galactus' answer. Let a = (2/5)t. Then
$\displaystyle a \left [ \begin{array}{c} 1 \\ \frac{5}{2} \end{array} \right ] = \frac{2}{5} t \left [ \begin{array}{c} 1 \\ \frac{5}{2} \end{array} \right ] = t \left [ \begin{array}{c} \frac{2}{5} \\ 1 \end{array} \right ] $
which is the same as galactus' eigenvector. So the two are essentially the same.
-Dan