Chapter 6---Section 1 Eigenvalues and Eigenvectors
Chapter 6 Eigenvalues
§1 Eigenvalues and Eigenvectors
New words and phrase
Eigenvalue 特征值
Eigenvector 特征向量
Diagonalization 对角化
Characteristic polynomial 特征多项式l
Eigenspace 特征空间
1.1 Definitions
A motivation for diagonalization of matrices: choosing a basis so that the matrix representation of a linear transformation becomes easier.
★Definition Let A be an nxn matrix. A scalar is said to be an eigenvalue or a characteristic value of A if there exists a nonzero vector x such that Ax=x. The vector x is said to be a eigenvector or a characteristic vector belonging to .
Characteristic polynomial of A:
Let A be an nxn matrix and be a scalar. The following statements are equivalent:
(a) is an eigenvalues of A.
(b) has a nontrivial solution.
(c)
(d) is singular.
(e)
Example Let . Find the eigenvalues and the corresponding eigenvectors.
Solution . Thus the characteristic polynomial has roots . Solve the system for the eigenspace corresponding to each .
1.2 Complex Eigenvalues
If A is an nxn matrix with real entries, then the characteristic polynomial of A will have real coefficients, and hence all its complex roots must occur in conjugate pairs. Thus if =a+bi () is an eigenvalue of A, then =a-bi is must also be an eigenvalue of A. If is a matrix with complex entries, then . .
Not only do the complex eigenvalues of a real matrix occur in conjugate pairs, but so do the complex eigenvectors. Indeed, if is a complex eigenvalue of a real nxn matrix A and z is an eigenvector belonging to , then is an eigenvector belonging to .
1.3 The Product and Sum of the Eigenvalues
(1) The characteristic polynomial of A is a polynomial of degree n. The and terms are contained only in the product
Remark: Expanding along the first column,
=…..
The minors , i=2, 3, …, n, do not contain the two diagonal elements and . Expanding in the same manner, we conclude that is the only term in the expansion is the only term involving a product of more than n-2 of the diagonal elements.
(2) The product of the eigenvalues of A equals det(A)
, det(A)=p(0)
(3) The sum of the eigenvalues of A equals the trace of A (the sum of diagonal entries of A),
tr(A)=
Remark: Compare the coefficient of in and , we obtain the results.
1.4 Similar Matrices
Theorem 6.1.1 Let A and B be nxn matrices. If B is similar to A, then the two matrices have the same characteristic polynomial and consequently both have the same eigenvalues.Chapter 6---Section 3 Diagonalization
§3 Diagonalization
New words and phrases
Diagonalization 对角化
Diagonalizability 可对角化性
Diagonalizable 可对角化的
3.1 Conditions for Diagonalizability
Theorem 6.3.1 If are distinct eigenvalues of an nxn matrix A with corresponding eigenvectors , then are linearly independent.★Definition An nxn matrix is said to be diagonalizable if there exists a nonsingular matrix X and a diagonal matrix D such that
we say that X diagonalizes A.
Theorem 6.3.2 An nxn matrix A is diagonalizable if and only if A has n linearly independent eigenvectors.Proof
=XD
Remarks
1. If a is diagonalizable, then the column vectors of the diagonalizing matrix X are eigenvectors of A, and the diagonal elements of D are the corresponding eigenvalues of A.
2. The diagonalizing matrix X is not unique. Reordering the columns of a given diagonalizing matrix X or multiplying them by nonzero scalars will produce a new diagonalizing matrix.
3. If A is nxn an A has distinct eigenvalues, then A is diagonalizable. If the eigenvalues are not distinct, then A may or may not be diagonalizable depending on whether A has n linearly independent eigenvectors.
4. If A is diagonalizable, then a can be factored into a product XDX-1
Example: Diagonalizing the matrix
Eigenvalues are 1, -1, 3 and -3
Corresponding eigenvectors are
, ,,
AX=(X
X-1AX=diag(1, -1, 3, -3)
3.2 The Exponential of a Matrix
(Remark: This part is optional.)
Consider a system of linear differential equations
The system can be written as
Y’=AY
If is an eigenvalue of A and x is an eigenvector belonging to , then is a solution of the system Y’=AY. This will be true whether is real or complex.
Given a scalar x, the exponential can be expressed in terms of a power series
Similarly, for any nxn matrix A, we can define the matrix exponential in
The matrix exponential occurs in a wide variety of applications. In the case of a diagonal matrix
If , then
If , then
Example Solving the equation .
It is more difficult to compute the matrix exponential for a general nxn matrix A. If, however, A is diagonalizable, then
for k=1, 2, ….
The matrix exponential can be applied to the initial value problem
Since the expansion of has infinite radius of convergence, we have
If we set
Then . Thus, the solution to the initial value problem is simply .
3.3 Diagonalization of Symmetric Matrices
Theorem （Corolllary 6.4.3 on page 350) The eigenvalues of a symmetric matrix are all real. Furthermore, eigenvectors belonging to distinct eigenvalues are orthogonal.Proof Let A be a symmetric matrix. Let be an eigenvalue of A and let x be an eigenvector belonging to . If , then
and its transpose is
. Thus, is real.
If y and z are vectors belonging to distinct eigenvalues and , then
. Thus
Theorem (Corollary 6.4.5, page 350) If A is real symmetric matrix, then there is an orthogonal matrix U that diagonalizes A, that is , where D is diagonal.Proof Let be an eigenvalue of A, and let be a unit eigenvector belonging to . Construct such that form an orthonormal basis for , thus is an orthogonal matrix
for
Then,
where M is an (n-1)x(n-1) symmetric matrix. By induction, there is an
orthogonal matrix such that is diagonal.
Let
Q=,
then
Explore the relationship between the dimension of an eigenspace corresponding an eigenvalue and the multiplicity of the eigenvalue
Let , be an eigenvalue of A whose eigenspace has dimension k, where 1<k<n. Any basis for the eigenspace can be extended to a basis for . Let and , then
Hence, the dimension of the eigenspace corresponding to the eigenvalue is less than the multiplicity of .
If A is symmetric, let be an orthonormal basis for the eigenspace corresponding to the eigenvalue , and extend it to an orthonormal basis for , ,
, we will show that can not have as a root.
If is a root of , there is a vector u such that . Let v=, then
Since are linearly independent, are linearly independent. Hence, are linearly independent. This contradicts that the eigenspace corresponding to the eigenvalue is k.
Therefore, for a symmetric matrix A, the dimension of the eigenspace of equals the multiplicity of .
and have the same characteristic polynomial . The eigenspace of A corresponding to the eigenvalue 1 is two, The eigenspace of B corresponding to the eigenvalue 1 is one.
If are linearly independent eigenvectors belonging to the eigenvalue s, are linearly independent eigenvectors belonging to the eigenvalue t (s is not equal to t), then , are linearly independent.
Proof If =0,
then =0. This implies that
, and . Hence, , are linearly independent.
Using induction, we can generalize this result.
If is a basis for the eigenspace corresponding to for i=1, 2, …, t, then the collection of those vector are linearly independent.
Chapter 6---Section 6 Quadratic Forms
§6 Quadratic Forms
New words and phrases
Quadratic form 二次型
Conic section 圆锥截面
Ellipse 椭圆
Parabola 抛物线
Parabolic 抛物的
Hyperbola 双曲线
Hyperbolic 双曲的
Cone 锥面
Ellipsoid 椭球面
Hyperboloid of one sheet 单叶双曲面
Hyperboloid of two sheets 双叶双曲面
Elliptic paraboloid 椭圆抛物面
Hyperbolic paraboloid 双曲抛物面
Positive definite 正定的
Negative definite 负定的
Positive semidefinite 正半定的
Negative semidefinite 负半定的
Indefinite 不定的
6.1 Quadratic Forms in Two Variables
★Definition (Quadratic Equation in Two Variables)
A quadratic equation in two variables x and y is an equation of the form
(1)
The equation (1) can be written in the form
Let
The term = is called the quadratic form associated with (1).
The graph of an equation of the form (1) is called a conic [kanik] section.
If there are no ordered pairs (x, y) that satisfy (1), we say that the equation represents an imaginary conic. If the graph of (1) consists of a single point, a line, or a pair of lines, we say that (1) represents a degenerate conic. Graphs of nondegenerate conics turn out to be circles, ellipses, parabolas, or hyperbolas.
Standard Forms of Conics
(i) (a circle)
(ii) (an ellipse)
(iii) (a hyperbola)
(iv) (a parabola)
A conic is said to be in standard position if its equation can be put into one of these four standard forms.
There is little problem for transforming a quadratic equation into the standard form if xy term does not appear in the equation.
If, however, the xy term does appear, the xy term has to be eliminated by using a change of coordinates in order to obtain the standard form.
6.2 Quadratic Forms in n Variables
★Definition A quadratic equation (二次方程) in n variables is one of the form
,
where , A is an nxn symmetric matrix, B is a 1xn matrix, and is a scalar.
The vector form
is the quadratic form （二次型） in n variables associated with the quadratic equation.
The graph of a quadratic equation in three variables is called a quadratic surface（二次曲面）. There are four basic types of nondegenerate quadratic surfaces:
Ellipsoids （椭球面）
Spheres （球面）
Hyperboloids(of one sheet or two sheets).（双曲面（单叶或双叶））
Cones（锥面）
Paraboloids （Elliptic or Hyperbolic）（抛物面）(椭圆的或双曲的)）
Theorem 6.6.1 (Principal Axes Theorem) If A is real symmetric nxn matrix, then there is a change of variables such that , where D is a diagonal matrix.Understanding the geometric meaning of this theorem:
With respect to the standard basis ,
With respect to the new orthonormal basis, .
The change of coordinates corresponding to the change of basis from to is given by u=.
This theorem is a corollary of the theorem that a symmetric matrix is diagonalizable by an orthogonal matrix .
★Definition A quadratic form is said to be definite if it takes on only one sign as x varies over all nonzero vectors in .
The form is positive definite （正定的）if for all nonzero x in and negative definite （负定的）if for all nonzero x in .
A quadratic form is said to be indefinite（不定的） if it takes on values that differ in sign.
If and assumes the values 0 for some , then f(x) is said to be positive semidefinite.（正半定的）
If and assumes the values 0 for some , then f(x) is said to be negative semidefinite. （负半定的）
§7 Positive Definite Matrices
★Definition A real symmetric matrix A is said to be
I. Positive definite if for all nonzero x in .
II. Negative definite if for all nonzero x in .
III. Positive semidefinite if for all nonzero x in .
IV. Negative semidefinite if for all nonzero x in .
V. Indefinite if takes on values that differ in sign.
Properties of Positive Definite Matrices
Theorem 6.6.2 Let A be a real symmetric nxn matrix. Then A is positive definite if and only all its eigenvalues are positive.Proof: If A is positive definite and is an eigenvalue of A, then for any eigenvector x belonging to , , hence .
Suppose that all the eigenvalues of A are positive. Let {} be an orthonormal set of eigenvectors of A. If x is any nonzero vector in , then , where for i=1, 2, …, n.
Let A be a real symmetric nxn matrix. Then A is negaitive definite if and only all its eigenvalues are negative.
Facts:
I. If A is a symmetric positive definite matrix, then A is nonsingular. (this is because that all eigenvalues are all positive, and det(A)=the product of all eigenvalues.)
II. If A is a symmetric positive definite matrix, then det(A)>0
(this is because that all eigenvalues are all positive, and det(A)=the product of all eigenvalues.)
Leading principal submatrices
Given an nxn matrix A, let denote the matrix formed by deleting the last n-r rows and columns of A. is called the leading principal submatrix of A of order r.
Equivalent Conditions for Positive Definite Matrices
Theorem 6.7.1 Let A be a symmetric nxn matrix. The following are equivalent.(a) A is positive definite
(b) All eigenvalues of A are positive.
(c) The leading principal submatrices all have positive determinants.
(d) There is an orthogonal matrix Q such that , where D is diagonal and has positive diagonal entries.
(e) A can be factored into a product for some nonsingular matrix B.
(The proof of this theorem is required.)
Proof (a) çè (b) by Theorem 6.6.2.
(a)è(c): To show that is positive definite, Let be any nonzero vector in and set . Since , it follows that is positive definite, and consequently .