I have the following scaled matrix
$A = \frac{2}{3\sqrt{10}}
\begin{pmatrix}
5 & 0 \\
3 & 4
\end{pmatrix}$
Now I have to calculate the eigenvectors and values of $B = A^TA$
I'd like to make my life easier and get a better understanding of the influence of scaling on eigenvectors and values.
I defined $A' =
\begin{pmatrix}
5 & 0 \\
3 & 4
\end{pmatrix}, B' = A'^TA'$ and $z = (\frac{2}{3\sqrt{10}})^2 = \frac{4}{90}$
and figured that I can do: $B = zB' = z(A'^TA')$
Now my questions is if I can calculate the eigenvectors and values of $B'$ and then later scale it by $z$ or do I have to do $B = zB'$ and then calucate the eigenvectors and values of $B$
Best Answer
Go back to the fundamental definition of an eigenvector: it’s a vector $\mathbf v$ such that $M\mathbf v=\lambda\mathbf v$ for some scalar $\lambda$. If $\mathbf v$ is an eigenvector of $M$, then $$(cM)\mathbf v = c(M\mathbf v)=c(\lambda\mathbf v)=(c\lambda)\mathbf v,$$ that is, $\mathbf v$ is also an eigenvector of the “scaled matrix” $cM$ with eigenvalue $c\lambda$.