Foundations — Systems of first-order linear ODEs — matrix method

This page builds every symbol the parent note leans on, starting from nothing. Read top to bottom; each block earns the notation the next block uses. See the parent: Systems of first-order linear ODEs — matrix method.

1. What a derivative is (the prime mark ${}^{\prime }$)

Look at the figure: the curve is the value $y(t)$; the straight amber line just kissing it is the tangent, and its steepness is $y^{\prime }$. If $y$ is going up, $y^{\prime }>0$; flat means $y^{\prime }=0$; falling means $y^{\prime }<0$.

2. The scalar equation $y^{\prime }=ay$ and $e^{at}$

That self-copying property is exactly the sentence $y^{\prime }=ay$. So $y=C{e}^{at}$ solves it, where $C$ is the starting value $y(0)$.

3. Vectors — the bold $\mathbf{x}$ and its components

When $\mathbf{x}$ depends on time, $\mathbf{x}(t)$ is a moving arrow — its tip traces a path. Then ${\mathbf{x}}^{\prime }(t)$ is another arrow: the velocity, showing which way the tip is heading and how fast.

4. Matrices — the letter $A$ and $A\mathbf{v}$

For a $2\times 2$ example, $A\mathbf{v}=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{c}{v}_1\\ v_2\end{array}\right)=\left(\begin{array}{c}a{v}_1+b{v}_2\\ c{v}_1+d{v}_2\end{array}\right).$

In the figure a grey input arrow becomes a cyan output arrow — usually pointing somewhere else. That "somewhere else" is the whole difficulty: $A$ mixes the coordinates together.

5. Eigenvectors and eigenvalues — the special directions

Now the payoff of §4. Most arrows get twisted by $A$. But a few special arrows come out pointing the same way — only longer or shorter.

In the figure, the amber arrow is an eigenvector: after $A$ acts, it still lies on the dashed line — just rescaled. The grey arrow is not special — it swings off its line.

6. The identity $I$, singularity, and $\det$

To find eigenvalues we rewrite $A\mathbf{v}=\lambda \mathbf{v}$ as $(A-\lambda I)\mathbf{v}=0$. Three symbols appear:

The equation $(A-\lambda I)\mathbf{v}=0$ needs a nonzero $\mathbf{v}$ getting crushed to $0$ — that happens exactly when $\det (A-\lambda I)=0$. That is the characteristic equation.

7. Complex numbers $i$, $\alpha \pm i\beta$, and rotation

Sometimes the characteristic equation has no real roots — its solutions involve $i$.

Euler's formula (see Euler's Formula) ties this to rotation: $e^{(\alpha +i\beta )t}=e^{\alpha t}(\cos \beta t+i\sin \beta t).$ Read it as: $e^{\alpha t}$ sets the size (grow if $\alpha >0$, decay if $\alpha <0$), and $\cos \beta t,\sin \beta t$ spin it around a circle at rate $\beta$.

8. Constants $c_i$, superposition, and initial conditions

How the foundations feed the topic

Equipment checklist

Cover the right side and answer aloud — if any stumps you, reread that section.