Where can I calculate the exponential of a matrix online?

Question

Where can I exponentiate a $3\times 3$ matrix like $\left[\begin{array}{ccc} 0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 0 \end{array}\right]$ online?

Is there some website where this is possible?

Thank you very much.

Answer 1

Depending on which matrix exponential you want, you can use:

• Wolfram Alpha 1st option: $e^{A}$
• Wolfram Alpha 2nd option: $e^{At}$

This is actually a command in Mathematica.

I would be surprised if were not available in other CAS programs and some of those are online, like Sage, Maxima and others.

Answer 2

Somewhat perversely, could I suggest you first try analytically, by Sylvester's formula, or, more specifically, simpler methods as below?

Still, in your specific case, this would be overkill: your matrix is trivially diagonalizable, $$M=\left[\begin{array}{ccc} 0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 0 \end{array}\right]=\frac{1}{\sqrt2}\left[\begin{array}{ccc} 1 & 1 & 0\\ 1 & -1 & 0\\ 0 & 0 & {\sqrt2} \end{array}\right] ~~ \left[\begin{array}{ccc} 1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & 0 \end{array}\right]~\frac{1}{\sqrt2}\left[\begin{array}{ccc} 1 & 1 & 0\\ 1 & -1 & 0\\ 0 & 0 & {\sqrt2} \end{array}\right]\equiv R^{-1} D R,$$ for the obviously defined diagonal matrix D and diagonalizing rotations R.

It then immediately follows that $$ \exp M= R^{-1} e^D R= \left[\begin{array}{ccc} \cosh 1 & \sinh 1 & 0\\ \sinh 1 & \cosh 1 & 0\\ 0 & 0 & 1 \end{array}\right]. $$

Sylvester's formula generalizes to non-diagonalizable matrices, but my sense is you are dealing with diagonalizable ones.