Determining all vectors $v\in \mathbb{R}^3$ such that $[v]_C=[v]_B$

Question

Let $B=(u_1,u_2,u_3)$ and $C=(v_1,v_2,v_3)$ be two ordered bases fot $(\mathbb{R}^3:\mathbb{R}).$ If the transition matrix from $B$ to $C$ is

$P=\begin{bmatrix} 1 & 1 & -1 \\ -1 & 1 & 1 \\ 1 & -1 & 1 \end{bmatrix}$

then determine all vectors $v\in \mathbb{R}^3$ such that $[v]_C=[v]_B.$

I completely stack on this question.

Answer 1

The transition matrix $P$ does the following: $[v]_C=P[v]_B$. So if you want $[v]_C=[v]_B$, you're looking for a vector satisfying $[v]_B=P[v]_B$, or equivalently $[v]_C=P[v]_C$.

Explicitly, you're looking for solutions $[x,y,z]^T$ to the linear system: $$\begin{bmatrix} 1 & 1 & -1 \\ -1 & 1 & 1 \\ 1 & -1 & 1 \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix}=\begin{bmatrix} x \\ y \\ z \end{bmatrix}$$

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In other words: you're looking for fix points of $P$, vectors that do not change when multiplied by $P$.

In other words again: you're looking for an eigenvector of $P$, corresponding to eigenvalue $1$.