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I'm a uni student and having a bit of trouble with a homework question in an engineering "Optimisation" class. I have a matrix $N$, where they ask you to find the eigenvalues and eigenvectors, which I have done.

Then they ask for you to plot the eigenvalue ($\lambda$), the eigenvector ($z$) and then $\lambda * z$ on an Argand diagram.

I know how to plot the eigenvalue, but am having trouble with plotting the eigenvector (being a $2 \times 1$ matrix).

Say the eigenvector is $[(2 + 3i)/5 \, , 1]^{T}$, how would I plot this on an Argand diagram? What does it even mean to plot a matrix on an Argand diagram? Sorry for my lack of formatting, I am not very familiar with LaTeX. Thanks for any help you can provide! :)

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up vote 0 down vote accepted

What they probably mean is to take the two elements of your vector

$$\mathbf v = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = \begin{pmatrix} \frac{2 + 3i}{5} \\ 1 \end{pmatrix}$$

and plot the elements individually as vectors in the complex plane. That is, you will have one vector to the point $v_1 = \frac{2+3i}{5}$ and one to $v_2 = 1$. Then mark where your eigenvalue (lets say it's $\lambda = \lambda_x + i\lambda_y$) is. Then multiply your vector by the eigenvalue, which yields a new vector with elements $w_1 = (\lambda_x + i\lambda_y) \frac{2 + 3i}{5}$ and $w_2 = (\lambda_x + i\lambda_y)1$ and mark these in the complex plane as well.

It will probably be easier if you use polar form (at least if your eigenvalue is not real).

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Thanks very much for your help! – Rachel Mar 20 '11 at 8:14

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