7.3 FUNCTIONS OF DIAGONALIZABLE MATRICES

For square matrices A, what should it mean to write sin A, e ^A, ln A, etc.? A naive approach might be to simply apply the given function to each entry of A such as

(7.3.1)

But doing so results in matrix functions that fail to have the same properties as their scalar counterparts. For example, since sin ² x + cos ² x = 1 for all scalars x, we would like our definitions of sin A and cos A to result in the analogous matrix identity sin ² A + cos ² A = I for all square matrices A . The entrywise approach (7.3.1) clearly fails in this regard.

One way to define matrix functions possessing properties consistent with their scalar counterparts is to use infinite series expansions. For example, consider the exponential function

(7.3.2)

Formally replacing the scalar argument z by a square matrix A ( z ⁰ = 1 is replaced with A ⁰ = I ) results in the infinite series of matrices

(7.3.3)

called the matrix exponential. While this results in a matrix that has properties analogous to its scalar counterpart, it suffers from the fact that convergence must be dealt with, and then there is the problem of describing the entries in the limit. These issues are handled by deriving a closed form expression for (7.3.3).

If