8.1. Introduction to Eigenvalues and Eigenvectors¶
Matrix multiplication is like yoga to a vector — mostly stretching and
rotation. That is, when we multiply a matrix and a vector together
() the result is usually a stretching and a rotation of
the original vector . However, that is not always the case.
There are special matrices that rotate a vector, but do not change its length.
(See Rotation of a Point for more on rotation matrices.) There are also vectors
corresponding to each square matrix that will be stretched by the
multiplication, but not rotated. This later case is what we consider here.
Consider the following matrix and three vectors.
Fig. 8.1 shows the three vectors and the
product of each vector with
matrix . The product shows the
typical result of matrix to vector multiplication. The multiplication both
stretches and rotates the vector. But in the cases of
and , the vectors are
stretched, but not rotated. This is because vectors and
are eigenvectors to matrix . The eigenvalues
tell us the stretching (scaling) factor for each eigenvector.
Fig. 8.1 Eigenvectors are only scaled by matrix multiplication. Other vectors
are both scaled and rotated by multiplication.¶
When a vector is one of the eigenvectors of a matrix
, the following special relationship holds.
For the square matrix, there are eigenvectors,
, and corresponding scalar eigenvalues,
.
The power of this relationship is that when, and only when, we multiply a
matrix by one of its eigenvectors, then in a simplified equation we can
replace a matrix with just a simple scalar value. This simplification
provides an elegant solution to several otherwise complex problems.