Question

Need help understanding this concept !!!
http://prntscr.com/52nnag

Answer 1

@amistre64

Answer 2

A^(12) eh ... ill be back in an hour gotta go

Answer 3

@ganeshie8

Answer 4

@ParthKohli

Answer 5

\[A^k = S\Lambda^k S^{-1}\]

Answer 6

familiar with finding eigen values and eigen vectors ?

Answer 7

yes that is next unit.

Answer 8

what i dont understand from the link i posted above is that what is the role of the matrix A in computing the A^12

Answer 9

yes

Answer 10

but the link i posted is before the topic of eignevalue / eigenvector

Answer 11

start by finding the eigen values and eigen vectors

Answer 12

we can think about how to use them afterwards

Answer 13

http://prntscr.com/52nwuh

Answer 14

Oh, i knw only the method using eigen vectors for finding the power of a matrix

Answer 15

@Zarkon

Answer 16

the matrix on left is the eigen vector matrix
and the right matrix is the inverse of it

Answer 17

hmmm

Answer 18

if you already know finding eigen vectors and taking inverses, its pretty easy

Answer 19

here is an example
http://math.stackexchange.com/questions/597602/finding-a-2x2-matrix-raised-to-the-power-of-1000

Answer 20

ok so i understand from the picture i posted that the extreme left matrix is the eigenvector of A and the extreme right is the inverse of the eigenvector ?

Answer 21

@ganeshie8

Answer 22

yes, extreme left is the "eigen vector matrix"

Answer 23

if you solve for eigen values and find eigen vectors, you will get two eigen vectors

Answer 24

put those two vectors as columns in the matrix S

Answer 25

thats the left side matrix

Answer 26

the right most matrix is the inverse of S

Answer 27

http://www.wolframalpha.com/input/?i=eigen+vectors+%7B%7B0%2C-2%7D%2C%7B1%2C3%7D%7D

Answer 28

eigen vectors = \(\large \begin{pmatrix}-1\\1\end{pmatrix}\) and \(\large \begin{pmatrix}-2\\1\end{pmatrix}\)

Answer 29

the eigen vector matrix would be
\[\large S = \begin{pmatrix}-1&2\\1&1\end{pmatrix}\]

Answer 30

fill the diagonal matrix with eigen values :\[\large \Lambda = \begin{pmatrix}2&0\\0&1\end{pmatrix}\]

Answer 31

take the inverse of S and put it on right side

Answer 32

\[\large S^{-1} = \begin{pmatrix}1&2\\-1&-1\end{pmatrix}\]

Answer 33

\[\large \begin{align} A &= \begin{pmatrix}1&2\\-1&-1\end{pmatrix}\\~\\
&=S\Lambda S^{-1}\\~\\
&=\begin{pmatrix}-1&-2\\1&1\end{pmatrix}\begin{pmatrix}2&0\\0&1\end{pmatrix}\begin{pmatrix}1&2\\-1&-1\end{pmatrix}



\end{align}\]

Answer 34

\[\large \begin{align} A^k &=S\Lambda^k S^{-1}\\~\\
&=\begin{pmatrix}-1&-2\\1&1\end{pmatrix}\begin{pmatrix}2^k&0\\0&1^k\end{pmatrix}\begin{pmatrix}1&2\\-1&-1\end{pmatrix}



\end{align}\]

Answer 35

ahhhhhhhhh it all makes sense now. Thank you very much for helping me kind sir !

Answer 36

np :)