Basic Props. of a Fund. Mat. to a Lin. Homo. Sys. of First Order ODEs

Basic Properties of a Fundamental Matrix to a Linear Homogeneous System of First Order ODEs

Recall from The Determinant of a Fundamental Matrix to a Linear Homogeneous System of First Order ODEs page that if $\Phi$ is a solution to the matrix equation $X' = A(t)X$ on an interval $J = (a, b)$ then for any $\tau \in J$ and for all $t \in J$ we have that:

(1)

\begin{align} \quad \det \Phi (t) = \det \Phi (\tau) \cdot \mathrm{exp} \left ( \int_{\tau}^{t} \mathrm{tr} (A(s)) \: ds \right ) \end{align}

We noted that by Abel's Fundamental Matrix Formula theorem that if $\Phi$ is a fundamental matrix to the linear homogeneous system of first order ODEs $\mathbf{x}' = A(t) \mathbf{x}$ then $\Phi$ is a solution to $X' = A(t) X$, and so $\Phi$ satisfies the conditions above.

We now state some basic properties of fundamental matrices from these results.

Corollary 1: If $\Phi$ is a fundamental matrix to the linear homogeneous system of first order ODEs $\mathbf{x}' = A(t)\mathbf{x}$ and if for some $\tau \in J$, $\det \Phi (\tau) = 0$ then $\det \Phi (t) = 0$ for all $t \in J$.

• Proof: If for some $\tau \in J$, $\det \Phi (\tau) = 0$ then from the determinant formula for $\det \Phi(t)$ we have that for all $t \in J$:

(2)

\begin{align} \quad \det \Phi(t) &= \det \Phi (\tau) \cdot \mathrm{exp} \left ( \int_{\tau}^{t} \mathrm{tr} (A(s)) \: ds \right ) \\ &= 0 \cdot \mathrm{exp} \left ( \int_{\tau}^{t} \mathrm{tr} (A(s)) \: ds \right ) \\ &= 0 \quad \blacksquare \end{align}

Corollary 2: If $\Phi$ is a fundamental matrix to the linear homogeneous system of first order ODEs $\mathbf{x}' = A(t) \mathbf{x}$ and if for some $\tau \in J$, $\det \Phi (\tau) \neq 0$ then $\det \Phi(t) \neq 0$ for all $t \in J$.

• Proof: If for some $\tau \in J$, $\det \Phi (\tau) \neq 0$ then from the determinant formula for $\det \Phi (t)$ we have that for all $t \in J$:

(3)

\begin{align} \quad \det \Phi(t) &= \underbrace{\det \Phi (\tau)}_{\neq 0} \cdot \underbrace{\mathrm{exp} \left ( \int_{\tau}^{t} \mathrm{tr} (A(s)) \: ds \right )}_{>0} \\ \end{align}

• Therefore $\det \Phi(t) \neq 0$ for all $t \in J$. $\blacksquare$