Eigenvalues and Eigenvectors.
Definition: The real number  is said to be an eigenvalue of the n x n matrix A provided that
there exists a nonzero vector v such that
Av = v.
The vector v is called the eigenvector of the matrix A associated with the eigenvalue .
Eigenvalues and eigenvectors are also called characteristic values and characteristic vectors.
The equation | A - I | = 0 is called the characteristic equation of the square matrix A.
Theorem: The real number  is an eigenvalue of the n x n matrix A if and only if  satisfies the
characteristic equation.
The eigenspace associated with a fixed eigenvalue  is the solution space of the homogeneous
system (A - I ) v = 0.
Definition: The n x n matrices A and B are similar provided that there exists an invertible matrix
P such
B  P 1 AP
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
D  P 1 AP
Theorem: The n x n matrix A is diagonalizable if and only if it has n linearly independent
eigenvectors.
Theorem: The k eigenvectors v 1 , v 2 ,..., v k associated with the distinct eigenvalues 1 , 2 ,..., k
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 1 , 2 ,..., k be the distinct eigenvalues of the n x n matrix A. For each
i  1,2,...k , let S i be a basis for the eigenspace associated with i . Then the union S of the basis
 S i is a linearly independent set of eigenvectors of A.
Theorem: Eigenvectors associated with distinct eigenvalues of a symmetric matrix are
orthogonal.
Theorem: The following properties of square matrix A are equivalent:
(a)
(b)
(c)
(d)
A is orthogonal
A T is orthogonal
The column vectors of A are orthonormal
The row vectors of A are orthonormal.
Definition: The square matrix A is called orthogonally diagonalizable provided there exists an
orthogonal matrix P such that
D  P 1 AP ,
in which case
D  P T AP and A  PDP T
because P 1  P T
Theorem: The n x n matrix A is orthogonally diagonalizable if and only if it has n mutually
orthogonal eigenvectors.
Theorem: A square matrix is othogonally diagonalizable if and only if it is symmetric.
Theorem: The characteris equation of a symmetric matrix has only real solutions.
Gram-Schmidt Orthogonalization:
To replace the linearly independent vectors v 1 , v 2 ,..., v n one by one with mutually orthogonal
vectors u1 , u 2 ,..., u n that span the same subspace , begin with
u1  v 1 .
For k  1,2,...n  1 in turn, take
u v
u v
u v
u k 1  v k 1  1 k 1 u 1  2 k 1 u 2    k k 1 u k
u 1  .u 1
u 2  .u 2
u k  .u k