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Linear Algebra

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Distinct Eigenspaces of a Symmetric Matrix are Orthogonal

LINALG-2DMCDB

Consider the following matrix:

$$A=\begin{bmatrix}2&1&-1\\\ 1&2&-1\\\ -1 & -1& 2\end{bmatrix}$$

Then:

$$\mathbb{R}^3 = V_1 \oplus V_2$$

...where $V_1, $ and $V_2$ are two non-trivial orthogonal subspaces of $\mathbb{R}^3$ that are invariant under $A$, i.e., $A V_1=V_1$ and $A V_2=V_2$.

Which of the following are possibilities for $V_1$ and $V_2$ ?

A

$V_1=span\{\cfrac{1}{\sqrt{2}}\begin{bmatrix} -1\\\ 1\\\ 0\end{bmatrix}, \frac{1}{2\sqrt{6}}\begin{bmatrix} 1\\\ 1\\\ 2\end{bmatrix}\}$

and

$V_2=span\{\cfrac{1}{\sqrt{3}}\begin{bmatrix} 1\\\ 1\\\ -1\end{bmatrix}\}$

B

$V_1=span\{\begin{bmatrix} 1\\\ 0\\\ 0\end{bmatrix}, \cfrac{1}{\sqrt{2}}\begin{bmatrix} 0\\\ 1\\\ 1\end{bmatrix}\}$

and

$V_2=span\{\cfrac{1}{\sqrt{2}}\begin{bmatrix} 0\\\ -1\\\ 1\end{bmatrix}\}$

C

$V_1=V_2=span\{\frac{1}{\sqrt{3}}\begin{bmatrix} 1\\\ 1\\\ -1\end{bmatrix}\}$