Matrices
are widely used in maths, physics and other sciences to represent
linear relations between many 'inputs' and many 'outputs'. The matrix
is an array of elements, a_ij, each of which gives the weight
of the jth input in its effect on the ith output.
In maths text books they appear in 'linear algebra'. Here I deal
only with square matrices, which are ones with the same number of
rows as there are columns.
A matrix
can be pictured as operating (by matrix multiplication) on a set
of input values (written in the form of a vector) to convert it
into set of outputs (also written in the form of a vector). Geometrically
this corresponds to the matrix stretching and rotating the input
vector into the output one. An NbyN square matrx can be thought
of as operating in an Ndimensional space. Each N by N matrix has
a set of N special directions in that space which have the distinguishing
property that they are not rotated by the matrix multiplication
operation. In general the input vector along each one of these directions
is stretched or shrunk or swapped in the sense it points, but not
otherwise rotated. These are the 'eigenvectors' and the scaling
stretch factors are the 'eigenvalues'. These words come from the
German language and mean 'characteristic' or 'own' since they effectively
characterise the action of the matrix.
I have
written two articles on matrices and their eigenvalues. One is about
numerical methods for finding the eigenvalues and eigenvectors,
especially for quite large matrices. The second is about the statistical
properties of the eigenvalues of matrices whose elements are random
numbers. Both are large fields of study. Indeed, random matrices
have been shown to model seemingly unrelated situations in maths
and physics, including the distribution of complex zeros of the
Riemann zeta function and the energy levels of heavy atomic nuclei.
Article
1: "Iterative numerical methods for real eigenvalues and
eigenvectors of matrices"
