Useful properties of matrix exponentials

Useful properties of matrix exponentials#

Let \(A\in\mathbb{R}^{n\times n}\). Below are some useful properties of the matrix exponential \(e^{At}\).

We cannot compute the matrix exponential entrywise#

Property: In general, \((e^{A t})_{i j} \neq e^{A_{i j} t}\).

Proof: Consider the matrix \(A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}.\) Compute its powers:

\[\begin{split} A^2 = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}, \qquad A^3 = \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix}, \qquad A^k = \begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix}. \end{split}\]

Using this pattern, from the power series of \(e^{A t}\) :

\[e^{A t}= I + t A + { t^2 A^2 }/{ 2! } + { t^3 A^3 }/{ 3! } + \cdots\]

we obtain

\[\begin{split}e^{A t}= \begin{bmatrix} 1 + t + { t^2 }/{ 2! } + { t^3 }/{ 3! } + \cdots & 0 + t + 2 { t^2 }/{ 2! } + 3 { t^3 }/{ 3! } + \cdots \\[8pt] 0 & 1 + t + { t^2 }/{ 2! } + { t^3 }/{ 3! } + \cdots \end{bmatrix}.\end{split}\]

Therefore:

\[\begin{split} e^{A t} = \begin{bmatrix} e^t & t e^t \\ 0 & e^t \end{bmatrix}. \end{split}\]

This shows that:

\[ (e^{A t})_{12}=te^t \neq e^{A_{12} t} = e^t, \]

so \(e^{A t}\) cannot be computed entrywise.

Derivative of \(e^{A t}\)#

Property: It holds that

\[ {d \, e^{A t}}/{dt} = A \, e^{A t} = e^{A t} A. \]

Proof: We start from the series definition:

\[ e^{A t} = I + tA + {t^2 A^2}/{2!} + {t^3 A^3}/{3!} + \cdots. \]

Differentiate term by term:

\[ {d \, e^{A t}}/{dt} = {d \over dt} \left( I + tA + {t^2 A^2}/{2!} + {t^3 A^3}/{3!} + \cdots \right). \]

This gives:

\[ = 0 + A + tA^2 + {t^2 A^3}/{2!} + \cdots. \]

Factor out \(A\):

\[ = A \left( I + tA + {t^2 A^2}/{2!} + \cdots \right). \]

Recognize the series:

\[ = A \, e^{A t} \]

(OR)

\[ = e^{A t} A \quad\text{(compare to the scalar case).} \]

Matrix exponential of the zero matrix#

Property: \(e^0=I\).

Proof: From the series definition,

\[e^{A t} = I + t A + {t^2 A^2}/{2!} + {t^3 A^3}/{3!} + \cdots\]

it is straightforward to see that \(e^0=I\).

When can we use the multiplicative property?#

Property: In general,

\[ e^{(A_1 + A_2) t} \neq e^{A_1 t} e^{A_2 t}. \]

However, \( e^{(A_1 + A_2) t} = e^{A_1 t} e^{A_2 t} = e^{A_2 t} e^{A_1 t} \) if and only if \(A_1 A_2 = A_2 A_1\) (commutative) or \(t=0\) (trivial).

Inverse of the matrix exponential#

Property: \((e^{At})^{-1}=e^{-At}\).

Proof: Note that \(A\cdot(-A)=(-A)\cdot A=-A^2\), meaning that \(A\) and \(-A\) commute. Therefore, from the previous fact:

\[ e^{A t} e^{-A t} = e^{-A t} e^{A t} = e^{(A-A)t} = e^0 = I. \]

Hence, for all \(A\) and \(t\), \(e^{A t}\) is invertible and

\[(e^{A t})^{-1} = e^{-A t}. \]