Eigenvalues and eigenvectors
Given a square matrix A, we can write the eigenvalue equation A⋅x = λ⋅x,
where the values of λ that satisfy the equation are known as the eigenvalues
of matrix A. For each value of λ, we can find, from the same equation,
values of x that satisfy the eigenvalue equation. These values of x are known
as the eigenvectors of matrix A. The eigenvalues equation can be written
also as (A – λ⋅I)x = 0.
This equation will have a non-trivial solution only if the matrix (A – λ⋅I) is
singular, i.e., if det(A – λ⋅I) = 0.
The last equation generates an algebraic equation involving a polynomial of
order n for a square matrix A
. The resulting equation is known as the
characteristic polynomial of matrix A. Solving the characteristic polynomial
produces the eigenvalues of the matrix.
The calculator provides a number of functions that provide information
regarding the eigenvalues and eigenvectors of a square matrix. Some of
these functions are located under the menu MATRICES/EIGEN activated
Function PCAR generates the characteristic polynomial of a square matrix
using the contents of variable VX (a CAS reserved variable, typically equal to
'X') as the unknown in the polynomial. For example, enter the following
matrix in ALG mode and find the characteristic equation using PCAR: