# How to calculate eigenvectors from an identity matrix with eigenvalue $= 1$?

When determining the eigenvectors from matrix $A$: $$\begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix}$$

I found the eigenvalue to be $\lambda = 1$

calculating $(A - 1*\lambda)$ gives me the matrix : $$\left[ \begin{array}{cc|c} 0&0&0\\ 0&0&0 \end{array} \right]$$

Which eigenvector would this produce?

• Compute the eigenvalues again. – Git Gud Mar 19 '16 at 22:19
• How did you find the eigenvalue $-1$? – J.-E. Pin Mar 19 '16 at 22:19
• What is the solution space for the system related to $\;\det(A-1I)\;$ . The whole space, right? – DonAntonio Mar 19 '16 at 22:23
• The matrix is diagonal.....so the eigenvectors and eigenvalues should be obvious – ClassicStyle Mar 19 '16 at 22:27
• @TylerHG Not to someone who is new to the subject. – Arthur Mar 19 '16 at 23:20

For $\lambda=1$, the Eigenvector equation is

$$0v=0.$$

This clearly means that any vector is a solution.

By the way, this can be found directly by identifying

$$Av=\lambda v$$ and $$Iv=v.$$

You made a small error when calculating your eigenvalues. If you solve for the the determinant equal to zero you end up with the following equation: $(1-\lambda)(1-\lambda) = 0$.

Solving this you get both eigenvalues of $\lambda_1 = \lambda_2 = 1$. You can see from this how a diagonal matrix greatly simplifies your calculations. In a diagonal matrix the diagonal terms are your eigenvalues. Try recalculcating your eigenvectors with these eigenvalues. You will get $[0,\ 1]^T$ and $[1,\ 0]^T$.

Hope this helps.

This is fairly obvious, and can be solved with a bit of intuition without even touching an equation

The basis vector $\vec{i}$ equals $[1,0]^T$ and the basis vector $\vec{j}$ equals $[0,1]^T$. This means this matrix $(\vec i,\vec j)$ is the identity matrix. Nothing is changed because no vector is changed; every (nonzero) vector is an eigenvector with an eigenvalue of 1.

(sorry, I don't know how to use mathJax- I'm just a 14yr-old learning linear algebra through the youtube channel 3blue1brown. they have an excellent tutorial that will give you a visual intuition for what's going on) Here is the playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab00

• Welcome to Math.SE. I've taken the liberty to use some MathJax and $\LaTeX$ formatting, and hope that it will encourage you to use similar formatting in future posts to make your points. See this introduction. – hardmath May 9 '18 at 15:01