Matrix $A\in \mathbb{R}^{4\times4}$ has eigenvectors $\bf{u_1,u_2,u_3,u_4}$ satisfying $\bf{Au_1=5u_1,Au_2=9u_2}$ & $\bf{Au_3=20u_3}$. Find $A\bf{w}$.

Question

Problem

The matrix $A \in \mathbb{R}^{4\times4}$ has eigenvectors $\bf{u_1,u_2,u_3,u_4}$ where $\bf{u_1}=\begin{pmatrix}1\\1\\0\\1\end{pmatrix}$, $\bf{u_2}=\begin{pmatrix}1\\1\\1\\1\end{pmatrix}$, $\bf{u_3}=\begin{pmatrix}1\\-1\\0\\1\end{pmatrix}$ satisfy $A\bf{u_1=5u_1}$, $A\bf{u_2=9u_2}$ and $A\bf{u_3=20u_3}$.
Calculate $A\bf{w}$ where $\bf{w}=\begin{pmatrix}13\\7\\12\\13\end{pmatrix}$

At first I thought I should use $A=PDP^{-1}$ , where $P$ is eigenvector matrix and $D$ is eigenvalue matrix. If I'm not mistaken, from the question the eigenvalues are $\lambda_1=5$, $\lambda_2=9$ and $\lambda_3=20$ right? But I don't have the $\bf{u_4}$ and $\lambda_4$.

Do I need to know all the eigenvalues and eigenvectors? Do I need to find these values to calculate $A$ or is there another method?

Answer 1

HINT

We can calculate $Aw$ only if we can find $a,b,c$ such that

$$w=a\cdot u_1+b\cdot u_2+c\cdot u_3$$

thus you can easily find that: $b=12, a+c=1, a-c=-5$

Answer 2

Here $w$ is not a linear combination of $u_1, u_2, u_3$ so with this imformation it is not possible to calculate $Aw$.