Eigenvalues and Eigenvectors.
Let
be an matrix. If there exists a non-zero vector such that for some scalar then
- The scalar
is called an eigenvalue of . - The vector
is called an eigenvector of .
Remark
- In general, given any
matrix, . A scalar is an eigenvalue of iff the homogeneous equation has a non-trivial solution . - The homogeneous equation has a non-trivial solution iff
is singular, which in turn is equivalent to . - The equation
is called the characteristic equation of the matrix . The solutions to the characteristic equation are the eigenvalues of . - For each eigenvalue
, we can find the corresponding eigenvector by solving the equation
Algebraic and Geometric Multiplicity
Suppose
is an matrix with eigenvalues , some of them may be repeated.
- If one of the
appears times as a root to the characteristic equation then we say has algebraic multiplicity . - If an eigenvalue
gives rise to linearly independent eigenvectors, then we say that has geometric multiplicity .
Homogeneous Linear Systems with Constant Coefficients
Suppose the
matrix has eigenvalues (some of them may be repeated) and linearly independent eigenvectors . Then the general solution to the homogeneous system is given by
where‘s are arbitrary constants.