Eigenvalues and Eigenvectors.
Definition: The real number l is said to be an eigenvalue of the n x n matrix A provided
that there exists a nonzero vector v such that
Av = lv.
The vector v is called the eigenvector of the matrix A associated with the eigenvalue l.
Eigenvalues and eigenvectors are also called characteristic values and characteristic
vectors.
The equation | A - lI | = 0 is called the characteristic equation of the square matrix A.
Theorem: The real number l is an eigenvalue of the n x n matrix A if and only if l
satisfies the characteristic equation.
The eigenspace associated with a fixed eigenvalue l is the solution space of the
homogeneous system (A - lI ) v = 0.
Definition: The n x n matrices A and B are similar provided that there exists an invertible
matrix P such
APPB
1
Definition: The n x n matrix A is called diagonalizable if it is similar to a diagonal matrix
D, i.e., that is there exists a diagonal matrix D and an invertible matrix P such that
APPD
1
Theorem: The n x n matrix A is diagonalizable if and only if it has n linearly independent
eigenvectors.
Theorem: The k eigenvectors
k
v,...,v,v
21
associated with the distinct eigenvalues
k
lll
,...,,
21
of a matrix A are linearly independent.
Theorem: If the n x n matrix A has n distinct eigenvalues, then it is diagonalizable.
Theorem: Let
k
lll
,...,,
21
be the distinct eigenvalues of the n x n matrix A. For each
ki ,...2,1
, let
i
S
be a basis for the eigenspace associated with
i
l
. Then the union S of
the basis
i
S
is a linearly independent set of eigenvectors of A.
Theorem: Eigenvectors associated with distinct eigenvalues of a symmetric matrix are
orthogonal.
