# Linear Algebra : Eigenvalues and Eigenvectors

## Example Questions

### Example Question #51 : Eigenvalues And Eigenvectors

real matrix has as two of its eigenvalues 2 and . Give its characteristic equation.

Insufficient information is provided to answer the question.

Explanation:

A  matrix will have three (not necessarily distinct) eigenvalues, which are the zeroes of its characteristic polynomial equation. Since all of the entries of the matrix are real, all of the coefficients will be real as well. It follows that any imaginary zeroes must occur in conjugate pairs, so, since  is a zero of this equation, so is its complex conjugate,

The characteristic equation of a  equation sets a degree-3 polynomial equal to 0. Since 2, , and  are its zeroes, this polynomial is

,

The characteristic equation is

.

### Example Question #52 : Eigenvalues And Eigenvectors

is a  matrix;  is an eigenvalue of , with eigenspace of dimension 2.  and  are two eigenvectors of  corresponding to .

Does  exist so that  is also an eigenvector of  corresponding to ? If so, what is ?

No such  exists.

Explanation:

The two eigenvectors given,  and , are linearly independent, since they are not scalar multiples of each other; therefore, they form a basis of the 2-dimensional eigenspace of  is an eigenvector corresponding to  if and only if it is a linear combination of the two given basis vectors, or, equivalently, if there exist  so that

Rewrite this as follows:

This is equivalent to a system of three linear equations in two variables:

Solve the system by first, rewriting the second equation as

The first equation becomes

Substitute in the first equation:

Now substitute for  and  in the third equation:

, the correct response.

### Example Question #53 : Eigenvalues And Eigenvectors

is a  matrix;  is an eigenvalue whose eigenspace has dimension 2.   and  are two eigenvectors of  corresponding to ..

True or false:  is an eigenvector of  corresponding to .

True

False

False

Explanation:

The two eigenvectors given,  and , are linearly independent, since they are not scalar multiples; since the eigenspace corresponding to  has dimension 2, they form a basis of the eigenspace.  is an eigenvector corresponding to  if and only if it is a linear combination of the two given basis vectors, so we seek to find  so that

This is

This is the system of four linear equations in two variables:

If we look at the first and fourth equations, we can immediately see that this system is inconsistent. There is no  that solves this system. It follows that  is not an eigenvector of  corresponding to .

### Example Question #51 : Eigenvalues And Eigenvectors

is a nonsingular real matrix with four eigenvalues: .

True or false:  must have these same four eigenvalues.

False

True

False

Explanation:

One property of eigenvalues is that if  is nonsingular, the set of eigenvalues of  is exactly the set of reciprocals of eigenvalues of . The eigenvalues of  are , so the eigenvalues of these numbers are the reciprocals of these - in order, . This is not the same set.

### Example Question #55 : Eigenvalues And Eigenvectors

Give the set of eigenvalues of ; if an eigenvalue has multiplicity greater than 1, repeat the eigenvalue that many times.

Explanation:

is an eigenvalue of  if it is a solution of the characteristic polynomial equation

Set this equation:

The determinant can be found most easily by adding the products of each entry in one of the rows or columns to its corresponding cofactor. Since the second row has only one nonzero entry, we use this one:

the minor formed by striking out row 2 and column 2:

Take the upper-left to lower-right product, and subtract the upper-right to lower-left product:

Thus,

,

and

This makes  and 1 the solutions of the characteristic equation, the latter with multiplicity 2. It follows that  has as its eigenvalues the set .

### Example Question #56 : Eigenvalues And Eigenvectors

is a  matrix. One of its eigenvalues is 2, with corresponding eigenvector . The other eigenvalue is 3, with corresponding eigenvector .

Find .

Insufficient information is given to answer the question.

Explanation:

Let  for some complex values of the variables.

is an eigenvector of  corresponding to eigenvalue  if

.

is an eigenvector corresponding to eigenvalue 2, so

and

is an eigenvector corresponding to eigenvalue 2, so, similarly,

Two systems of two linear equations in two variables are created. Each can be solved separately.

Multiplying the bottom equation by 2, then adding:

The second system is

Solve similarly:

### Example Question #52 : Eigenvalues And Eigenvectors

Complete the theorem by filling in the blank: Let  be an  matrix.  is an eigenvalue of  if and only if ____________.

Explanation:

This expression is equivalent to saying "find the characteristic polynomial of , and set it equal to ." This is the most frequently used method of finding eigenvalues of a matrix.

### Example Question #51 : Eigenvalues And Eigenvectors

True or false; If  is an eigenvalue of a matrix , then  is invertible.

True

False

False

Explanation:

For example,, has  as one of its eigenvalues, but is clearly not invertible since its determinant is .

### Example Question #54 : Eigenvalues And Eigenvectors

is an eigenvalue of a nonsingular real matrix .

True or false: It follows that  is an eigenvalue of .

True

False

True

Explanation:

The eigenvalues of a matrix  are the solutions of the characteristic polynomial equation

.

is a matrix with only real entries, so the coefficients of the polynomial must be real as well; it follows that any imaginary solutions are in conjugate pairs. , an eigenvalue of , is a solution of the characteristic equation; consequently, its complex conjugate  is a solution, and thus, an eigenvalue of .

One property of eigenvalues is that if  is nonsingular, the set of eigenvalues of  is exactly the set of reciprocals of eigenvalues of . Since  is an eigenvalue of , it follows that  is an eigenvalue of .

### Example Question #52 : Eigenvalues And Eigenvectors

Is 1 an eigenvalue of , and if so, which is an eigenvector of 1?

(You might find a calculator with matrix arithmetic capability helpful.)

1 is an eigenvalue of  with eigenvectors  and .

1 is not an eigenvalue of .

1 is an eigenvalue of  with eigenvector .

1 is an eigenvalue of  with eigenvector .

1 is an eigenvalue of  with eigenvector .

1 is an eigenvalue of  with eigenvector .

Explanation:

While the question can be answered by direct definition, it is arguably easier to answer it by noting that the columns of  each have entries adding up to 1 - the characteristics of a stochastic matrix; each column has the same four entries, and

.

A matrix with this property has eigenvalue 1 - in fact, it is the dominant eigenvalue.

An eigenvector of  corresponding to an eigenvalue  is that vector  such that

.

Set ; this becomes

Since  is the dominant eigenvalue, as is raised to a large enough power,  approaches  the desired eigenvector. Finding, say, , this vector approaches

.

Multiplying this by 4, it can easily been seen that   is an eigenvector of  corresponding to eigenvalue 1.