Eigenvalues and Eigenvectors

The previous chapters have all been asking “what does this transformation do to space”. This chapter asks a different question: are there special vectors whose direction is unchanged by the transformation? The answer is yes — they are the eigenvectors, and the scaling factor is the eigenvalue.

The value of this viewpoint is that it translates “matrix operations” into “the essential structure of a transformation”. The hardest part of a matrix is that it stretches every direction to a different length and rotates every direction to a different angle; but eigenvectors tell us that along certain directions the transformation does nothing but pure scaling. The eigenvalues characterise those scaling factors, and hence characterise the overall tendency of the transformation to “stretch or compress”.

This chapter starts from the basic concepts of eigenvalues and eigenvectors and uses visualisation to let you see directly that “some directions are preserved by the transformation”; it then covers similarity and diagonalisation — if a matrix can be diagonalised, the original complicated transformation becomes a set of independent scalings along the coordinate axes that do not interfere with one another, and the amount of computation drops sharply; finally it discusses real symmetric matrices, whose special properties guarantee that all eigenvalues are real and that eigenvectors are mutually orthogonal, which is the reason real symmetric matrices are so important in physics and numerical computation.