Hello everyone, this nxn matrix arises in my numerical scheme for solving a diffusion PDE.(adsbygoogle = window.adsbygoogle || []).push({});

[itex]

M =

\left(\begin{array}{cccccccccc}1-\frac{Dk}{Vh} & \frac{Dk}{Vh} & 0 & 0 & & & \ldots & & & 0

\\[6pt] \frac{Dk}{h^2} & 1-2\frac{Dk}{h^2} & \frac{Dk}{h^2} & 0 & & & & & &

\\[6pt]0 & \frac{Dk}{h^2} & 1-2\frac{Dk}{h^2} &\frac{Dk}{h^2} & & & & & &

\\

\\

\\\vdots & & & & \ddots & & & & & \vdots

\\

\\

\\

\\[6pt] & & & & & & \frac{Dk}{h^2} & 1-2\frac{Dk}{h^2} & \frac{Dk}{h^2} & 0

\\[6pt] & & & & & & & \frac{Dk}{h^2} & 1-2\frac{Dk}{h^2} & \frac{Dk}{h^2}

\\[6pt] 0 &\ldots & & & & & & & 2\frac{Dk}{h^2} & 1-2\frac{Dk}{h^2}

\end{array}\right)

[/itex]

I can easily use Gershgorin disks, and the freedom to set a constraint between h and k to guarantee that all eigen vectors are between -1 and 1, or even between 0 and 1 if that is more advantageous.

I need to prove I can diagonalize this matrix, so I am attempting to show that there are n linearly independent eigen vectors.

I considered trying to show that the eigen vectors are distinct but I am really not sure where to start.

I seem to remember one method being something like guessing [itex]λ = cos(2\pi/n)[/itex] but I don't see how to go form there.

Any suggestions? I have lots of experience in PDE's and ODE's but have no formal linear algebra experience, just what I have taught myself.

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# Linearly independent eigen vectors

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