Chapter 6 Eigenvalues

Example In a certain town, 30 percent of the married women get divorced each year and 20 percent of the single women get married each year. There are 8000 married women and 2000 single women, and the total population remains constant. Let us investigate the long-range prospects if these percentage of marriages and divorces continue indefinitely into the future.

1 Eigenvalues and Eigenvectors Definition Let A be an n×n matrix. A scalar is said to be an eigenvalue or a characteristic value of A if there exists a nonzero vector x such that. The vector x is said to be an eigenvector or a characteristic vector belonging to. Example Let and

The subspace N( A - I ) is called the eigenspace corresponding to the eigenvalue. The polynomial is called the characteristic polynomial, and equation is called the characteristic equation for the matrix A.

Let A be an n×n matrix and be a scalar. The following statements are equivalent: (a) is an eigenvalue of A. (b) has a nontrivial solution. (c) (d) is singular. (e)

Example Let Example Find the eigenvalues and the corresponding eigenvectors of the matrix Find the eigenvalues and the corresponding eigenspaces.

The Product and Sum of the Eigenvalues Expanding along the first column, we get

The sum of the diagonal elements of A is called the trace of A and is denoted by tr(A). Example If

Some Properties of the Eigenvalues: 1. Let A be a nonsingular matrix and let be an eigenvalue of A, then is an eigenvalue of A Let be an eigenvalue of A and let x be an eigenvector belonging to, then is an eigenvalue of and x is an eigenvector of belonging to for m=1, 2, …. 3. Let, and let be an eigenvalue of A, then is an eigenvalue of.

Example If the eigenvalues of matrix A are: 2, 1, -1, then find the eigenvalues for the following matrices: (a) (b)

Similar Matrices Theorem Let A and B be n×n matrices. If B is similar to A, then the two matrices both have the same characteristic polynomial and consequently both have the same eigenvalues.

3 Diagonalization Theorem If are distinct eigenvalues of an n×n matrix A with corresponding eigenvectors x 1, x 2, …,x k, then x 1, …, x k are linearly independent. Definition An n×n matrix A is said to be diagonalizable if there exists a nonsingular matrix X and a diagonal matrix D such that X -1 AX=D We say that X diagonalizes A.

Theorem An n×n matrix A is diagonalizable if and only if A has n linearly independent eigenvectors. 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 an n×n matrix and A has n 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 Let Determine whether the matrix is diagonalizable or not.

Definition If an n×n matrix A has fewer than n linearly independent eigenvectors, we say that A is defective. Theorem If A is an n×n matrix and are s distinct eigenvalues for A, let be the basis of, where, then are linearly independent.

Example Let Determine whether the two matrices are diagonalizable or not. and

Some Results for Real Symmetric Matrix: 2. If are distinct eigenvalues of an n×n real symmetric matrix A with corresponding eigenvectors x 1, x 2, …,x k, then x 1, …, x k are orthogonal. 3. If A is a real symmetric matrix, then there is an orthogonal matrix U that diagonalizes A, that is, U -1 AU=U T AU=D, where D is diagonal. 1. The eigenvalues of a real symmetric matrix are all real.

Example Let Find an orthogonal matrix U that diagonalizes A. Example Let Find an orthogonal matrix U that diagonalizes A.

6 Quadratic Forms Definition A quadratic equation in two variables x and y is an equation of the form (1) Equation (1) may be rewritten in the form (2) Let The term is called the quadratic form associated with (1). and

Conic Sections The graph of an equation of the form (1) is called a conic section. A conic section is said to be in standard position if its equation can be put into one of these four standard forms: or (circle) (ellipse) (hyperbola) (parabola)

Example Consider the conic section This equation can be written in the form The matrixhas eigenvaluesand with corresponding unit eigenvectorsand

Let and set Thus and the equation of the conic becomes

Quadratic Surfaces (ellipsoid)(cone) (hyperboloid of one sheet)(hyperboloid of two sheets) (elliptic paraboloid)(hyperbolic paraboloid)

A quadratic form including n variables is ：

Theorem For any quadratic form X T AX, we can find an orthogonal transformation X=CY such that Y T BY is in standard form. Example Let Find an orthogonal transformation X=CY such that Y T BY is in Standard form.

Example For the conic section Find an orthogonal transformation X=CY such that Y T BY is in standard form.

Definition A quadratic form f(x)=x T Ax is said to be definite if it takes on only one sign as x varies over all nonzero vectors in R n. The form is positive definite if x T Ax>0 for all nonzero x in R n and negative definite if x T Ax<0 for all nonzero x in R n. A quadratic form is said to be indefinite if it takes on values that differ in sign. If f(x)=x T Ax ≥0 and assumes the value 0 for some x≠0, then f(x) is said to be positive semidefinite. If f(x) ≤0 and assumes the value 0 for some x≠0, then f(x) is said to be negative semidefinite.

Definition A real symmetric matrix A is said to be I. Positive definite if x T Ax>0 for all nonzero x in R n. Ⅱ. Negative definite if x T Ax<0 for all nonzero x in R n. III. Positive semidefinite if x T Ax≥0 for all nonzero x in R n. IV. Negative semidefinite if x T Ax≤0 for all nonzero x in R n. V. Indefinite if x T Ax takes on values that differ in sign.

Theorem Let A be a real symmetric n×n matrix. Then A is positive Definite if and only if all its eigenvalues are positive. Theorem Let A be a real symmetric n×n matrix. Then A is positive definite if the leading principal submatrices A 1, A 2, …, A n of A are all positive definite.

Example Determine whether the quadratic form is positive definite.