Matrices – Matrix Exponential of Non-Diagonalizable Matrix

matricesmatrix exponential

I am currently self-learning about matrix exponential and found that determining the matrix of a diagonalizable matrix is pretty straight forward :).

I do not, however, know how to find the exponential matrix of a non-diagonalizable matrix.

For example, consider the matrix
$$\begin{bmatrix}1 & 0 \\ 1 & 1\end{bmatrix}$$

Is there any process of finding the exponential matrix of a non-diagonalizable matrix? If so, can someone please show me an example of the process? :). I am not looking for an answer of the above mentioned matrix (since I just made it up), but rather I'm interested in the actual method of finding the matrix exponential to apply to other examples 🙂

Best Answer

There are two facts that are usually used for this computation:

Theorem: Suppose that $A$ and $B$ commute (i.e. $AB = BA$). Then $\exp(A + B) = \exp(A)\exp(B)$

Theorem: Any (square) matrix $A$ can be written as $A = D + N$ where $D$ and $N$ are such that $D$ is diagonalizable, $N$ is nilpotent, and $ND = DN$

With that, we have enough information to compute the exponential of every matrix.

For your example, we have $$ D = \pmatrix{1&0\\0&1} = I, \quad N = \pmatrix{0&0\\1&0} $$ we find that $$ \exp(D) = eI\\ \exp(N) = I + N + \frac 12 N^2 + \cdots = I + N + 0 = I + N $$ So, we have $$ \exp(D + N) = \exp(D) \exp(N) = (eI)(I+N) = e(I+N) = \\ \pmatrix{e&0\\e&e} $$

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