UNI_FOUNDATION · Linear algebra · LA.EIGEN

Eigenvalues and eigenvectors

Intuition

Understand first, compress later.

Linear algebra describes linear structure using vectors, matrices and transformations.

Switch representations

Matrix as a transformationA = [1 0; 0 1], det A = 1

original gridimage grid under Aimage of the unit square, area 1î ↦ (1, 0)ĵ ↦ (0, 1)

A sends î to (1, 0) and ĵ to (0, 1); the faint original grid becomes the gold image grid. The unit square (area 1) becomes the parallelogram of area |det A| = 1 — the size of the determinant is exactly that area factor.

Formal view

You can eigenvalues and eigenvectors when you can recognise the underlying objects, select valid rules and check each step for equivalence or logical implication.

Why does it work?

This skill sits on a prerequisite network. If a step is unstable, repair the missing prerequisite instead of merely doing more items of the same type.