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Linear Algebra

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Orthogonality of Distinct Eigenspaces of a Symmetric Matrix

LINALG-Q0GUEN

Consider the following matrix:

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

If $u, w$ are non-zero vectors such that $Au=4u$ and $Aw=w$, then which of the following statements are true.

A

$u=w$

B

$u$ is orthogonal to $w$, i.e., $\langle u, w\rangle =0$

C

$4$ is a multiplicity $1$ eigenvalue of $A$ while $1$ is a multiplicity $2$ eigenvalue of $A$.

D

$u=-w$