University Mathematics Handbook (2015)

X. Algebra

Chapter 11. Eigenvalues and Eigenvectors

11.1  Definitions; Diagonalization of Matrices

a.  Non-zero vector is an eigenvector of matrix if it is proportionate (parallel) to vector , that is

Scalar is the eigenvalue corresponding to
eigenvector .

b.  Matrix is diagonizable if and only if has linearly independent eigenvectors. Furthermore, if is a matrix in which the columns are eigenvectors of , then, the diagonal of diagonal matrix will have the corresponding eigenvalues, when the order of diagonal elements in corresponds to the order of rows in .

11.2  Characteristic Polynomials

a.  Polynomial is a characteristic polynomial of and matrix is a characteristic matrix of .

b.  The roots of characteristic polynomial are the eigenvalues of matrix .

c.  The characteristic polynomial of a finite dimension linear operator is the characteristic polynomial of its representative matrix.

d.  Eigenvectors of matrix corresponding to different eigenvalues are linearly independent.

e.  If matrix has different eigenvalues, then is diagonizable.

11.3  Eigenvalues of Operators

Let be a linear operator.

a. Vector is an eigenvector of operator if and if there exists scalar , such that . Scalar is calledeigenvalue of corresponding to the eigenvector ( is an eigenvector corresponding to ).

b. The kernel of is called an eigenspace of corresponding to eigenvalue and is denoted .

c. is diagonizable if, and only if, has a basis constructed of eigenvectors of . In such a case, , when is the eigenvalue of corresponding to .

11.4  Invariant Subspaces

a.  Let is linear operator over vector space , and let be a subspace of and be a restriction of over , then subspace W is -invariant of if .

b.   and are invariant subspaces of under every linear operator in .

c.  If is an eigenvalue of then the corresponding eigenspace is -invariant.

d.  If is a -invariant subspace of and is a basis of , and is a basis of , then is a block triangular matrix where , , and is the matrix representing the restriction of over .

e.  If is a diagonizable operator over a finite-dimension vector space , are the eigenvalues of , and are the corresponding eigenspaces, then is the direct sum of the eigenspaces .

11.5  Geometric and Algebraic Multiplicity of Eigenvalues

a.  The geometric multiplicity of eigenvalue of operator is a dimension of the eigenspace of corresponding to .

b.  The algebraic multiplicity of eigenvalue of operator is the multiplicity of as a root of the characteristic polynomial .

c.  Matrix has geometric multiplicity of equal to .

d.  The geometric multiplicity of eigenvalue does not exceed the algebraic multiplicity of .

e.   is diagonizable if and only if every eigenvalue of has a geometric multiplicity equal to its algebraic multiplicity.

11.6  Cayly-Hamilton Theorem

a.  Let be a polynomial of degree with coefficients of field and . Then, the polynomial in

is a matrix of the following properties:

1. If then, for every polynomial there holds .

2. If there exists invertible matrix such that is a diagonal matrix, then, for that holds .

b.  Cayley-Hamilton Theorem: If is a square matrix and its characteristic polynomial, then .

c.  If is invertible, then is the polynomial on .

11.7  Minimal Polynomial

a.  The polynomial equation of the lowest degree which satisfies is called minimal polynomial and is denoted as .

b.  For every polynomial which satisfies , minimal polynomial is a divider of ((.

c.  Every square matrix has a unique minimal polynomial.

d.  Every root of the minimal polynomial is an eigenvalue.

e.   and have the same roots.

f. Similar matrices have the same minimal polynomial.

g.  Let be block diagonal matrix , where the main diagonal of has square matrices . Then is the least common product of the corresponding minimal polynomials .

11.8  Spectral Factorization

a.  Transformation is called the projection on parallel to if (see 8.4).

b.  Operator holding is called projection operator.

c.  Spectral Factorization Theorem: Let be a diagonizable operator over -dimensional vector space, . Let be various eigenvalues of , and let be he eigenspaces of , respectively, then:

1.  .

2.  If is the projection of parallel to the direct sum of the rest of eigenspaces, then . The representation is the spectral factorization of .

3.  For every polynomial ,

4.  , when (polynomials are called Lagrange Polynomials).

5.  A factorization where the are different, , for every , is unique.

11.9  Jordan Form

a.  Jordan block, , is a -order matrix where the diagonal elements equal to the elements above the diagonal parallel to it equal , and the rest of the coefficients equal .

b.  The characteristic polynomial and minimal polynomial of are . The algebraic multiplicity of on is , and its geometric multiplicity is .

c.  A matrix is of Jordan form if it is a block diagonal matrix where the diagonal cells are Jordan blocks.

d.  If can be reduced to triangular form, then is similar to a matrix of a Jordan form. This matrix is unique up to the order of blocks.

e.  If when are Jordan blocks, then, for every matrix similar to :

1.  

2.  For every eigenvalue , the power of in is the highest Jordan block order corresponding to .

3.  The geometric multiplicity of eigenvalue is the number of blocks corresponding to it.