Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler

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

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

The direction field has an arrow with slope at each point coming from the differential equation. Arrows with the same sl... Read more

Transform each term in the linear differential equation to create an algebra problem. You can transform the algebra solu... Read more

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

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

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

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

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

Solutions to second order equations can approach infinity or zero. Saddle points have a positive and also a negative exp... Read more

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

An eigenvalue / eigenvector pair leads to a solution to a constant coefficient system of differential equations. Combina... Read more

A matrix produces four subspaces: column space, row space (same dimension), the space of vectors perpendicular to all ro... Read more

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

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

Diagonalizing a matrix also diagonalizes all its powers.

Vectors are a basis for a subspace if their combinations span the whole subspace and are independent: no basis vector is... Read more

Imaginary exponents with pure oscillation provide a "center" in the phase plane. The point (position, velocity) travels ... Read more

The impulse response is the solution when the force is an impulse (a delta function). This also solves a null equation (... Read more

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