Finding the Null Space Error

Question

I am watching a video on SVD where the instructor is solving the the nullspace of

$$\begin{bmatrix}26 & 18\\\ 18 & 74\end{bmatrix}$$ wich has the Eigen values $\lambda_1 = 20$ and $\lambda_2 = 80$

When he solves for these I believe he made an error and I am struggling to see if I made the error. My answers are reversed: For $\lambda_1$ my Eigen vector is $\begin{bmatrix} 1/\sqrt{10} \\ -3/\sqrt{10} \end{bmatrix}$ and $\lambda_2$ my Eigen vector is $\begin{bmatrix} 3/\sqrt{10} \\ 1/\sqrt{10} \end{bmatrix}$

And so my overall Eigen matrix would be:

$$\begin{bmatrix} 1/\sqrt{10} && 3/\sqrt{10} \\ -3/\sqrt{10} && 1/\sqrt{10}\end{bmatrix}$$

But they get:

I am sure I am wrong, but where am I wrong?

EDIT:

To show my work on the first Eigen value of 20 we solve for:

$$\begin{bmatrix} 6 && 18 \\ 18 && 54 \end{bmatrix}$$

which gives

$$\begin{bmatrix} 1 && 3 \\ 0 && 0 \end{bmatrix}$$

Solving for $x_1$ and $x_2$ we get:

$$x_1 + 3x_2 = 0$$

Answer 1

Based on the update, we can see what’s going wrong with your calculations. From $x_1+3x_2=0$ we have $x_1=-3x_2$. Remember that in the equation $(A-\lambda I)\mathbf x=0$ that you’re solving $\mathbf x=(x_1,x_2)$. If you substitute for $x_1$ in this, you end up with $(-3x_2,x_2)$. You appear to have done this backwards and substituted $-3x_2$ for $x_2$ instead, perhaps because the term involved $x_2$.

Answer 2

Since non-zero scalar multiples of eigenvectors are eigenvectors with the same eigenvalue, you can see that $\begin{bmatrix}26 & 18\\\ 18 & 74\end{bmatrix}\left(\begin{matrix}1\\-3 \end{matrix}\right)\ne 20 \left(\begin{matrix}1\\-3 \end{matrix}\right).$

So your first vector isn't correct. However, $\left(\begin{matrix}1\\3 \end{matrix}\right),$ is correct. It appears you have a sign error from where you computed a basis for the kernel of $\ker\left(\begin{bmatrix}26 & 18\\\ 18 & 74\end{bmatrix} -\lambda I_2\right).$

I suspect you made a similar error on the other vector.

Most importantly though, you should know the order of your eigenvalues also determines the order of the basis of eigenvectors, so the order of your vectors being reversed means that you mixed up your eigenvectors, or your eigenvalues at some point in your computation.

For SVD, Jordan form, or diagonalization problems, it's crucial that you know that the order in which you choose your eigenvalues determines the order in which you must write the columns of your eigenmatrix (basis).