# Decomposition of Matrix Exponential

Jump to navigation
Jump to search

## Theorem

Let $\mathbf A$ be a square matrix of order $m$ for some $m \in \Z_{\ge 1}$.

Let $t \in \R$ be a real number.

Let $e^{\mathbf A t}$ denote the matrix exponential of $\mathbf A$.

Let $\mathbf P$ be a non-singular matrix of order $m$.

Then:

- $e^{\mathbf P \mathbf A \mathbf P^{-1} } = \mathbf P e^{\mathbf A} \mathbf P^{-1}$

This article, or a section of it, needs explaining.In particular: Review what is meant by $e^{\mathbf A}$. We have only got $e^{\mathbf A t}$ defined.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Explain}}` from the code. |

## Proof

$\paren {\mathbf P \mathbf A \mathbf P^{-1} }^n = \mathbf P \mathbf A^n \mathbf P^{-1}$ by induction.

This needs considerable tedious hard slog to complete it.In particular: The above needs to be shown.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Finish}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

This theorem requires a proof.In particular: The result follows from plugging in the matrices and factoring $\mathbf P$ and $\mathbf P^{-1}$ to their respective sides.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |