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The example has sine of a vector in it. I haven't had coffee yet today but I've never heard of being able to compute sine of a vector. What does this mean? Defining it by its Taylor series doesn't work like it does for square matrices because vectors can't be multiplied, and if you assume the direct product is what's meant then each term in the Taylor series is a different sized matrix and can't be added. Surely it doesn't mean elementwise sine?

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It is element-wise sin, which is not entirely correct. You can take the sin of, or any analytic function of a matrix, however you'd have to compute the characteristic polynomial for the matrix and apply the function to that. It is a result of the Cayley-Hamilton Theorem. here is a tutorial: http://web.mit.edu/2.151/www/Handouts/CayleyHamilton.pdf

You don't need to know the characteristic polynomial to do it. Just sum as many terms of the power series as you want.

What the characteristic polynomial lets you do is to calculate it faster and more accurately because very large powers can be rewritten in terms of smaller powers that you already computed.

Yes it's element-wise, it's defined in the sidebar tab "Operators". In general the syntax seems closely aligned with Matlab, where all the math functions are elementwise.

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