Functions Applied to Square Matrices

Question

You can always multiply two square matrices of the same size. If we multiplied a matrix by itself, that would be $M^2$. We could keep on multiplying it by itself to get arbitrarily large exponents. If you can raise matrices to powers, and multiply/divide by a real number, than can you have polynomials applied to matrices? If that is the case, then, through Taylor series, couldn't almost any function apply to square matrices? Does this have any meaningful interpretation?

Answer 1

Yes, you can apply polynomial functions to square matrices - for example, a square matrix satisfies its own characteristic polynomial.

You can also define an exponential function on square matrices, using the same Taylor series expansion as in real analysis. Proving that the Taylor series converges is a little more tricky than for real numbers, but is not too difficult. The exponential function on matrices is used in differential geometry.

Once you have the exponential function then you can define trigonometric and hyperbolic functions on square matrices too.

Answer 2

There is a whole book "Functions of Matrices: Theory and Computation" http://www.maths.manchester.ac.uk/~higham/fm/ by Nicholas J. Higham, with accompanying MATLAB Matrix Function Toolbox http://www.ma.man.ac.uk/~higham/mftoolbox/ .

The book includes a chapter on matrix logarithms. Matrix logarithms will serve you well as we boldly move into an ever more multivariate era.