Matrices Prove (Diagonal Matrices)

Question

Let $A$ and $P = \begin{bmatrix} u & v & w \end{bmatrix}$ be 3 × 3 matrices where $u$, $v$, $w$ are columns of $P$ such that $Au=au$, $Av=bv$ and $Aw=cw$ for some real numbers $a$, $b$ and $c$. Show that if $P$ is invertible, then

$A= P\begin{bmatrix}a & 0 & 0\\0 & b & 0 \\ 0 & 0 & c\end{bmatrix} P^{-1}$

I've done:

$\begin{bmatrix}u & v & w\end{bmatrix} \begin{bmatrix}a & 0 & 0\\0 & b & 0 \\ 0 & 0 & c\end{bmatrix} P^{-1}$

$= \begin{bmatrix}au & bv & cw\end{bmatrix} P^{-1} $

$= \begin{bmatrix}Au & Av & Aw\end{bmatrix} P^{-1} $

$= A \begin{bmatrix}u & v & w\end{bmatrix} P^{-1} $

$= A P P^{-1} $

$= A I $

$= A $ (Q.E.D.)

May i know does this constitute to the prove?

From your problem statement, it seems that $B$ and $P$ are the same thing. You also refer to a $C$. What is that? — Matthew Leingang, Sep 06 '16 at 15:30
OK, that's better. I still think $B$ is redundant and you probably mean $P$. Otherwise, your proof looks valid. You might prettify it by letting $D$ be the diagonal matrix in the middle, and showing $AP = PD$. — Matthew Leingang, Sep 06 '16 at 15:35

score 1 · Answer 1 · answered Sep 06 '16 at 19:19

Since the assumption here is that if $P$ is invertible,

($P$ is invertible) $\implies$ $A = P \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix} P^{-1}$

So by stating the LHS, as long as you can prove the RHS, then your proof is sound.

Matrices Prove (Diagonal Matrices)

1 Answers1