Matrix exponential

We begin with the properties that are immediate consequences of the definition as a power series:

One of the reasons for the importance of the matrix exponential is that it can be used to solve systems of linear ordinary differential equations. The solution of

The matrix exponential can also be used to solve the inhomogeneous equation

There is no closed-form solution for differential equations of the form

However, for matrices that do not commute the above equality does not necessarily hold.

In fact, this gives a one-parameter subgroup of the general linear group since

The coefficients in the expression above are different from what appears in the exponential. For a closed form, see derivative of the exponential map.

This result also allows one to exponentiate diagonalizable matrices. If

Application of Sylvester's formula yields the same result. (To see this, note that addition and multiplication, hence also exponentiation, of diagonal matrices is equivalent to element-wise addition and multiplication, and hence exponentiation; in particular, the "one-dimensional" exponentiation is felt element-wise for the diagonal case.)

Since the series has a finite number of steps, it is a matrix polynomial, which can be computed efficiently.

This means that we can compute the exponential of X by reducing to the previous two cases:

Therefore, we need only know how to compute the matrix exponential of a Jordan block. But each Jordan block is of the form

Thus, as indicated above, the matrix A having decomposed into the sum of two mutually commuting pieces, the traceful piece and the traceless piece,

The procedure is much shorter than Putzer's algorithm sometimes utilized in such cases.

From before, we already have the general solution to the homogeneous equation. Since the sum of the homogeneous and particular solutions give the general solution to the inhomogeneous problem, we now only need find the particular solution.