Having trouble plotting a specific function

Question

let $$A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 5 & 6 \\ 4 & 8 & 9 \\ \end{pmatrix}$$

I want to plot this function $\lambda\to||(\lambda I_3-A)^{-1}||$

and then point out the eigen values on the graph

My try : since $\sum\lambda = Tr(A) = 15$ and $\prod\lambda = det(A) = -3 $

I chose this interval $x= -20:0.1:20$ all the eigenvalues sould be within that interval

then tried this : plot(x,norm((x.*B-A)^(-1))) where B =eye(3). but an error popped.

Please I need some help.

Thank you.

In case $\lambda$ is a scalar, $\lambda-A$ is undefined. – Michael Hoppe Mar 07 '17 at 14:00 — Michael Hoppe, Mar 07 '17 at 14:00
@MichaelHoppe oh sorry.. it was a typo. please see my edit – the_firehawk Mar 07 '17 at 14:02 — the_firehawk, Mar 07 '17 at 14:02

score 1 · Answer 1 · answered Mar 07 '17 at 15:12

When $\lambda$ is an eigenvalue of $A$ the matrix $\lambda I-A$ becomes singular (for any $A$). Thus, the inverse of this matrix does not exist at these points. In practice, when $\lambda$ is near a eigenvalue, the inverse "blows up". Thus, the Matlab gives an error.

If you just plot the determinant of $(x I -A)$, you will notice the eigenvalues when the shifted matrix becomes zero. Since, you have access to Matlab, it is rather easy to compute the eigenvalues of $A$ to confirm your observations.

Warning: Numerical analysts normally do not like to compute determinants. Perhaps, you should also compute the smallest singular value (using the svd in Matlab) and visualise the two plots.

Having trouble plotting a specific function

1 Answers1