# How to get an eigenvector of a $3\times 3$ matrix that has first column and a row of zeros

I have the following matrix $$\begin{bmatrix} 1& 0& 0\\ 0& 1& 1\\ 0& 1& 1 \end{bmatrix}$$

First I got the eigenvalues which are $0$, $1$, $2$.

I tried to get the eigenvectors associated with the above eigenvalues but I cannot in case of the eigenvalue $1$ as I got the following matrix $$\begin{bmatrix} 0& 0& 0\\ 0& 0& 1\\ 0& 1& 0 \end{bmatrix}$$ So, how can I get an eigenvector for this matrix?

• Just by looking at the matrix one can see that $\left(1, \begin{bmatrix}1 \\ 0 \\ 0 \end{bmatrix}\right)$ is an eigenpair of the first matrix and the other eigenvalues $0$ and $2$ follow easily. – Git Gud Apr 16 '14 at 22:03
• I tried online matrix calculator for checking eigen values and was correct. – Eng .. Abdalmonem Apr 16 '14 at 22:06
• @Eng..Abdalmonem $0,2$ and $4$ are not the eigenvalues, sorry. – Git Gud Apr 16 '14 at 22:07
• Yes i am sorry , you are correct. – Eng .. Abdalmonem Apr 16 '14 at 22:09
• It's a pair in which the first entry is an eigenvalue and the second is an eigenvector associated to the first entry. – Git Gud Apr 16 '14 at 22:12

The RREF of $[A-1I]v_1=0$ is:

$$\begin{bmatrix}0 & 1 & 0\\0 & 0 & 1 \\ 0 & 0 & 0\end{bmatrix}v_1 = 0$$

What if we chose:

$$v_1 = (1,0,0)$$

Update If we write:

$$0a + 1b + 0c = 0 \\ 0 a + 0b + 1 c = 0 \\ 0 a + 0 b + 0c = 0$$

What choices will actually make all three equations true and not be a zero eigenvector? What if we choose $(a , b , c) = (1, 0 , 0)$? Substitute those values back in and see if it satisfies the system.

Do you see how that satisfies the system? Recall that you cannot have a zero eigenvector. Would any other choice for $b = 1$, or $c=1$ or $b = c = 1$ work? No.

Also, see how this is the null space of the RREF?

• No I do not , can you explain a bit more ? – Eng .. Abdalmonem Apr 16 '14 at 22:07
• and how you got v1=(1,0,0) ? – Eng .. Abdalmonem Apr 16 '14 at 22:07
• Judging from your "last seen" (on this site) datum, you were up WAY TOO LATE last night! ;-) – Namaste Apr 17 '14 at 11:29
• I've done the same. I actually function best (both when I was a student) and onward, when I take a long "nap" (at the end of the work day), then awaken refreshed and working late. (Early rising, late bedtime - but a nice long nap in between!) – Namaste Apr 17 '14 at 12:35