It arises because .The same thing is of course true of . Another example is The inverse is The identity matrix may be expanded in terms of the eigenvectors of the matrix A . Before illustrating some more complicated functions let us see what it takes to prove Sylvester's theorem. Clearly, there is nothing peculiar about matrices either. Since almost any function can be made up of infinite series, we can consider also transcendental functions like sine, cosine, exponential.
Exponentials arise naturally as the solutions to differential equations.
If the matrix A is one of that vast majority which can be diagonalized, then the exponential can be more simply expressed by Sylvester's theorem. Boundary conditions are then used to determine the numerical values of k 1 and k 2 . An interesting situation arises with the square root of a matrix. A matrix like A will have four square roots because there are four possible combinations for choice of plus or minus signs on and .In general, an matrix has 2 n square roots. The best way to visualize the square root of this differential operator is to relate it to the square root of the matrix M where The right-hand matrix is a second difference approximation to a second partial derivative.