- #1
Jonnyb302
- 22
- 0
Hello everyone, this nxn matrix arises in my numerical scheme for solving a diffusion PDE.
[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.
[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.