---
tag: 0MQ24RBG93
kind: definition
name: Fundamental matrix
msc: [34A30]
defines: 0H1GN6P9R5
uses: [0A6QZ367GX, 0H4KNNCHPY]
sources:
  - url: https://en.wikipedia.org/wiki/Fundamental_matrix_(linear_differential_equation)
    locator: lead section
---

## Definition

Let $I$ be an open interval and let $A : I \to \mathbb{R}^{n \times n}$ be continuous.
A fundamental matrix of the homogeneous [[linear system|0A6QZ367GX]] $x' = A(t) x$ is a differentiable function $\Phi : I \to \mathbb{R}^{n \times n}$ with these two properties.
For each $t \in I$, $\Phi'(t) = A(t)\, \Phi(t)$.
For each $t \in I$, the matrix $\Phi(t)$ is invertible.

## Remarks

Equivalently, the columns of $\Phi$ are $n$ solutions that form a basis of the space of solutions.
Each solution is $x(t) = \Phi(t)\, c$ for a unique $c \in \mathbb{R}^n$, and the solution with $x(t_0) = x_0$ is $\Phi(t) \Phi(t_0)^{-1} x_0$.
If $\Phi'(t) = A(t) \Phi(t)$ and $\Phi(t_0)$ is invertible at one point $t_0$, then $\Phi(t)$ is invertible at each point, by [[Liouville's formula|09PS3S0EF6]].
For a constant matrix $A$, $\Phi(t) = e^{tA}$ is a fundamental matrix. See [[the matrix exponential as the solution|0YC8JT513J]].

## Explanations

### L1 human

A fundamental matrix puts $n$ independent solutions side by side as columns.
Then one matrix product gives each solution: choose the vector $c$ of coefficients.