Question

**Will Medal**

Answer 1

use your calculator

Answer 2

without calculator

Answer 3

do you know how to find eigen values and eigen vectors? Do you know what diagonalization is ?

Answer 4

there are other ways, so I need to know what you are doing in class if not.

Answer 5

yes isn't it A=SDS^-1?

Answer 6

Correct. But you first need to find the Eigen values and vectors so that you can construct \(S\)

Answer 7

ok so after finding eigen values and vectors what would i do to find the matrix?

Answer 8

Ok if we call your 2x2 matrix given above A
and the matrix of eigen vectors S. Then you will find D by

D = \(SAS^{-1}\). Now once you have D you just take the square root of the entries (they will only be non 0 on the diagonal). You with me?

Answer 9

oh ok i see, I would get a diagonal matrix and then I can just square root the diagonal is that correct?

Answer 10

then \(\sqrt{A} = S^{-1}\sqrt{D}S\)

Answer 11

yes

Answer 12

ok perfect thank you!

Answer 13

np

Answer 14

\[\left[\begin{matrix}4 & 3 \\ 2 & 3\end{matrix}\right]=\left[\begin{matrix}-1 & \frac{3}{2} \\ 1 & 1\end{matrix}\right]\times \left[\begin{matrix}1 & 0 \\ 0 & 6 \end{matrix}\right] \times \left[\begin{matrix}\frac{-2}{5} & \frac{3}{5} \\ \frac{2}{5} & \frac{2}{5} \end{matrix}\right]=\\=S D S^{-1}\\\sqrt{A}==S \left[\begin{matrix}\sqrt{1} & 0 \\ 0 & \sqrt{6}\end{matrix}\right] S^{-1}\]