Function of this matrix

Question

Given a matrix M=$ \begin{bmatrix} 2 & 1 \\ 1 & 2 \\ \end{bmatrix}$

How do I find $ \cos ({Mπ \over 6})$

do you know how to perform a matrix exponential? – Exodd Feb 03 '17 at 17:59 — Exodd, Feb 03 '17 at 17:59

score 2 · Accepted Answer · answered Feb 03 '17 at 18:00

2

Hint: Begin by diagonalizing. We have $M = UDU^T$ where $$ D = \pmatrix{3&0\\0&1}, \quad U = \frac{1}{\sqrt{2}}\pmatrix{1&1\\1&-1} $$

answered Feb 03 '17 at 18:00

Ben Grossmann

225,327

This was useful! Does this work whenever I need to find function of a matrix? – Pruthvi Bharadwaj Feb 03 '17 at 18:11
It works so long as $M$ is diagonalizable. – Ben Grossmann Feb 03 '17 at 18:14

score 0 · Answer 2 · answered Feb 03 '17 at 18:09

I will give you an example and you can apply it to your case. To arrive to the concept of cosinus of a matrix you need to think about power series for the cosine and call the sum the cosine of A.

In general (if $A$ is a matrix, let's say 2x2):

$cos(A)=I_{2x2}-\frac{1}{2!}A^2+\frac{1}{4!}A^4-...$

(where $I_{2\times2}$ is the identity matrix)

So you can apply that to your matrix by putting $M$ instead of $A$ and after you can multiply the elements of the matrix by $\pi/6$.

I hope this will help you

Function of this matrix

2 Answers2