# E Raised to the Power of a Matrix

1. May 24, 2014

### kq6up

1. The problem statement, all variables and given/known data

Use a spectrally decomposed matrix and the power series to yield $$e^{M}$$ where $$M=\begin{pmatrix} 5 & -2 \\ -2 & 2 \end{pmatrix}$$.

2. Relevant equations

$${ M }^{ n }=C{ D }^{ n }{ C }^{ -1 }$$

3. The attempt at a solution

Is this correct? $${ e }^{ M }=\frac { 1 }{ 5 } \begin{pmatrix} 1 & 1 \\ 2 & \frac { -1 }{ 2 } \end{pmatrix}\left( \sum _{ i=0 }^{ \infty }{ \frac { \begin{pmatrix} { 1 }^{ n } & 0 \\ 0 & { 6 }^{ n } \end{pmatrix} }{ n! } } \right) \begin{pmatrix} 1 & 2 \\ 4 & -2 \end{pmatrix}$$

Thanks,
Chris

2. May 24, 2014

### D H

Staff Emeritus
Yes, that's correct, but you really should expand that out.

3. May 24, 2014

### kq6up

Is this better? $$CI{ C }^{ -1 }+CD{ C }^{ -1 }+\frac { 1 }{ 2 } C{ D }^{ 2 }{ C }^{ -1 }+\frac { 1 }{ 6 } C{ D }^{ 3 }{ C }^{ -1 }\dots$$.

Chris

4. May 24, 2014

### D H

Staff Emeritus
No. It's a 2x2 matrix. There should be four simple expressions, one for each element of the matrix.

5. May 24, 2014

### kq6up

There might be some small error in there because when I use == with $$Ce^{D}C^-1$$ in Sage, I get a false. I am taking that to mean they are not equivalent.

$${ e }^{ 7/2 }\left( \begin{array}{rr} \frac { 1 }{ 2 } \, \cosh \left( \frac { 5 }{ 2 } \right) +\frac { 3 }{ 10 } \, \sinh \left( \frac { 5 }{ 2 } \right) & -\frac { 2 }{ 5 } \, \sinh \left( \frac { 5 }{ 2 } \right) \\ -\frac { 2 }{ 5 } \, \sinh \left( \frac { 5 }{ 2 } \right) & \frac { 1 }{ 2 } \, \cosh \left( \frac { 5 }{ 2 } \right) -\frac { 3 }{ 10 } \, \sinh \left( \frac { 5 }{ 2 } \right) \end{array} \right)$$

Thanks,
Chris Maness

6. May 24, 2014

### D H

Staff Emeritus
You should be able to compute $\exp(D)$ by hand. You do not need a symbolic math processor for this. You don't need a symbolic math processor for any of this. It's a product of three 2x2 matrices.

7. May 24, 2014

### kq6up

I did it by hand. I was just checking my results on Sage.

Chris

8. May 24, 2014

### D H

Staff Emeritus
What did you do by hand? That matrix in post #5? How did you arrive at that result?

What did you calculate $\exp(D)$ to be?

9. May 24, 2014

### kq6up

There was another problem in the set that was similar, and I had the answer. So I knew that it would have sinh(x), cosh(x) in the answer because of this answer, so I just made it fit that form so I can use sinh(x) and cosh(x). It didn't come out as pretty I was hoping for.

Thanks,
Chris Maness

10. May 24, 2014

### D H

Staff Emeritus
Yes, you can express this result using the hyperbolic functions. You made an error somewhere, however.

If you solve the problem as suggested you will *not* be using hyperbolic functions. What did you get for $\exp(D)$? Did you even try doing that? That is part of what the question asks you to find. It is very simple result.

11. May 24, 2014

### kq6up

Yes, exp(D) is $$\begin{pmatrix} { e }^{ 1 } & 0 \\ 0 & { e }^{ 6 } \end{pmatrix}$$.

Chris

12. May 24, 2014

### kq6up

This is what I got before I tried to convert it to hyperbolic functions:

$$\left( \begin{array}{rr} \frac { 4 }{ 5 } \, e^{ 6 }+\frac { 1 }{ 5 } \, e & -\frac { 2 }{ 5 } \, e^{ 6 }+\frac { 2 }{ 5 } \, e \\ -\frac { 2 }{ 5 } \, e^{ 6 }+\frac { 2 }{ 5 } \, e & \frac { 1 }{ 5 } \, e^{ 6 }+\frac { 4 }{ 5 } \, e \end{array} \right)$$

I did it by hand and with Sage. I got the same result.

Thanks,
Chris Maness

13. May 24, 2014

### D H

Staff Emeritus
That's the correct result. You made some error in converting that to hyperbolic expressions.

14. May 25, 2014

### Ray Vickson

Why would you want to try to convert to hyperbolic functions? It would just make everything longer and messier. Isn't it easier to just write $e$ instead of $\sinh(1)+\cosh(1)?$