If A and B are "similar" then B has the same eigenvalues as A.

The eigenvectors remain in the same direction when multiplied by the matrix. Subtracting an eigenvalue from the diagonal leaves a singular matrix: determinant zero. An n by n matrix has n eigenvalues.

A matrix can be diagonalized if it has n independent eigenvectors. The diagonal matrix Λ is the eigenvalue matrix.

Diagonalizing a matrix also diagonalizes all its powers.

An eigenvalue / eigenvector pair leads to a solution to a constant coefficient system of differential equations. Combinations of those solutions lead to all solutions.

The incidence matrix has a row for every edge, containing -1 and +1 to show which two nodes are connected by that edge.

Capturing all combinations of the columns gives the column space of the matrix. It is a subspace (such as a plane).

Vectors are a basis for a subspace if their combinations span the whole subspace and are independent: no basis vector is a combination of the others. Dimension = number of basis vectors.

A matrix produces four subspaces: column space, row space (same dimension), the space of vectors perpendicular to all rows (the nullspace), and the space of vectors perpendicular to all columns.

A graph has nodes connected by edges (other edges can be missing). This is a useful model for the Internet, the brain, pipeline systems, and much more.

A critical point is a constant solution to the differential equation. The slope of the right hand side decides stability or instability.

With two equations, the two linearized equations use the 2 by 2 matrix of partial derivatives of the right hand sides.

Two equations with a constant matrix are stable (solutions approach zero) when the trace is negative and the determinant is positive.

A box in the air can rotate stably around its shortest and longest axes. Around the middle axis it tumbles wildly.

The direction field has an arrow with slope at each point coming from the differential equation. Arrows with the same slope lie along an "isocline".

Solutions to second order equations can approach infinity or zero. Saddle points have a positive and also a negative exponent or eigenvalue.

With constant coefficients in a differential equation, the basic solutions are exponentials. The exponent solves a simple equation.

The impulse response is the solution when the force is an impulse (a delta function). This also solves a null equation (no force) with a nonzero initial condition.

Imaginary exponents with pure oscillation provide a "center" in the phase plane. The point (position, velocity) travels forever around an ellipse.

A second order equation gives two first order equations. The matrix becomes a companion matrix (triangular).

Combine null solutions to find a particular solution for any right hand side. But it may involve a difficult integral.

Transform each term in the linear differential equation to create an algebra problem. You can transform the algebra solution back to the ODE solution.

When the input force is an impulse, the output is the impulse response. For all inputs the response is a "convolution" with the impulse response.

With constant coefficients and special forcing terms (powers of t, cosines/sines, exponentials), a particular solution has this same form.

This method is also successful for forces and solutions equal to polynomials times exponentials. Substitute into the equation!