Linear Algebra Eigenvalues from a geometric description

Question

Goodmorning,

I have a question related to Linear Algebra:

The line L through the origin and (1,2) is given. Furthermore we know that for any vector v the vector Av is given by the projection of v onto the line L. Can you determine the eigenvalues and eigenvectors of A (without determining the matrix A explicitly)? If not, write down the answer 'no'.

What I did is the following:

1) First determine the equation of the line. The line passes through points (0,0) and (1,2) so the directional coefficient is 2-1 =2. The line passes through the origin, which gives equation y=2x for the line.

2) Determine the matrix by the projection: (([x y] * [1 2])/ ([1 2] . [1 2]))[1 2] (note that . indicates a dot product and I wrote all values in between brackets in column form, but I don't know how to write the values in brackets below each other).

3) This gives ((x+2y)/5)[1 2]. Using e1=[1 0] and e2=[0 1] with x,y we get a matrix: [1/5 2/5] [2/5 4/5]

4) Next I used eigenvalues -1 and +1 without any good reasoning (don't really know which eigenvalues to use). The matrix becomes: [(1/5)-lambda (2/5)] [(2/5) (4/5)-lambda]

5) Substituting the value of -1 for lambda first, we can solve the matrix [(1/5)+1 (2/5) 0] [(2/5) (4/5)+1 0]

If we have a free variable in the solution we can determine the corresponding eigenvector. The same for the other eigenvalue. I have the question whether the method I've used above is correct? I expect that I'm doing something wrong. Which eigenvalues should I use? The question explicitly states:"without determining matrix A", but with my method I do determine matrix A. How can I best solve this problem? Thank you in advance for the replies. I apologies for the above notation of the formulas, but unfortunately I don't know how programs such as LaTeX work.

Rik

Answer 1

In the spirit of this question, you're not supposed to set up the projection matrix $A$. The question seems to be designed to check if you understand, from a geometrical point of view, what eigenvectors and eigenvalues are.

Recall that an eigenvector is a vector which is mapped to a scalar multiple of itself with the corresponding eigenvalue being precisely that scalar multiple. For an orthogonal projection (in the plane) onto a line: which vector(s) will be mapped to a scalar multiple of itself?

If you have no idea: draw a few projections onto this line to get a feeling.

Hoover over for a hint:

Look at two special case: what happens to vectors parallel to the line (i.e. along the line) and what happens to vectors perpendicular to the line; how are they mapped?