Problem 1

Is an eigenvalue of ? Why or why not?

Yes is an eigenvalue as it is a root of the characteristic polynomial.

Problem 2

Is an eigenvalue of ? Why or why not?

Yes is an eigenvalue as it satisfies the equation for eigenvalues.

Problem 3

Is an eigenvector of ? If so, find the eigenvalue.

No it is not as there is no which satisfies the equation.

Problem 4

Is an eigenvector of ? If so, find the eigenvalue.

Yes it is an eigenvector with eigenvalue

Problem 5

Is an eigenvector of ? If so, find the eigenvalue.

Yes it is an eigenvector with the eigenvalue .

Problem 6

Is an eigenvector of ? If so, find the eigenvalue.

Yes it is an eigenvector with eigenvalue

Problem 7

Is an eigenvalue of ? If so, find one corresponding eigenvector.

Yes it is an eigenvalue as it satisfies the definition of eigenvalue.

Problem 8

Is an eigenvalue of ? If so, find one corresponding eigenvector.

Yes it is an eigenvalue as it satisfies the definition of eigenvalue.

Problems 9

Find a basis for the eigenspace corresponding to each listed eigenvalue.

(a).

(b).

(c).

(d).

(e).

(f).

(g).

(h).

Problems 10

Find the eigenvalues of each matrix.

(a).

(b).

Problem 11

For , find one eigenvalue with no calculation. Justify your answer.

One eigenvalue is 0 because when we take the determinant of we will get some product of times something because the lower two rows will go to zero.

Problem 12

Without calculation, find one eigenvalue and two linearly independent eigenvectors of . Justify your answer.

One eigenvalue is 0 because when we take the determinant of we will get some product of times something because the lower two rows will go to zero.

Two linearly independent eigenvectors are and

Problem 13

is an matrix. Mark each statement True or False. Justify each answer.

(a). If for Some Vector , then is an Eigenvalue of

False via the definition of an eigenvalue can not be the 0 vector which is not a condition in the question.

(b). A Matrix is not Invertible if and only if 0 is an Eigenvalue of

True, If the matrix has the eigenvalue 0 it's determinant must be zero which is an equivalent condition to the matrix being invertible.

(c). A Number is an Eigenvalue of if and only if the Equation Has a Nontrivial Solution

True via the definition of an eigenvalue and because nontrivial solutions necessarily means

(d). Finding an Eigenvector of May Be Difficult, but Checking whether a given Vector is in Fact an Eigenvector is Easy

True as it just requires one matrix multiplication and some basic division.

(e). To Find the Eigenvalues of , Reduce to Echelon Form

False to find eigenvalues find the roots of for some parameter .

Problem 14

(a). If for Some Scalar , then is an Eigenvector of

False, might be the zero vector which contradicts the definition of an eigenvector.

(b). If and Are Linearly Independent Eigenvectors, then They Correspond to Distinct Eigenvalues

False linearly independent vectors can have the same eigenvalue if the eigenvalue has a multiplicity greater than 1 in the characteristic polynomial

(c). The Eigenvalues of a Matrix Are on Its Main Diagonal

False the eigenvalues of a matrix depend on it's determinant which can introduce additionally constants and factors. This is only true if forms a triangular matrix.

(d). An Eigenspace of is a Null Space of a Certain Matrix

True the eigenspace of is the null space of for it's eigenvalues

Problem 15

Explain why a matrix can have at most two distinct eigenvalues. Explain why an matrix can have at most distinct eigenvalues.

Because the maximum degree of the characteristic polynomial can only be equal to the number of diagonal entries in the original matrix matrices only have 2 diagonal entries so the maximum degree is 2 and thus it can only be factored to a maximum of two terms giving at max 2 distinct eigenvalues.

Problem 16

Construct an example of a matrix with only one distinct eigenvalue.

Problem 17

Let be an eigenvalue of an invertible matrix . Show that is an eigenvalue of . Hint: Suppose a nonzero satisfies .

Problem 18

Show that if is the zero matrix, then the only eigenvalue of is 0.

Problem 19

Show that is an eigenvalue of if and only if is an eigenvalue of . Hint: Find out how and are related.

Via the definition of the eigenvalue we can see the statement is true

Problem 20

Consider an matrix with the property that the row sums all equal the same number . Show that is an eigenvalue of . Hint: Find an eigenvector.

An eigenvector of the of the matrix is as it's matrix product with will just be multiplied by the sum of all it's rows which is .

Problem 21

Consider an matrix with the property that the column sums all equal the same number . Show that is an eigenvalue of .

Via problem 19 we know that the eigenvalues of and are equivalent so taking the transpose of gives us the same matrix as problem 20 which gives us the same proof.

Problems 22

Let be the matrix of the linear transformation . Without writing , find an eigenvalue of and describe the eigenspace.

(a). Is the Transformation on that Reflects Points across Some line through the Origin

An eigenvalue is and it's eigenspace is all vectors normal to the line that points are reflected across which start at the origin.

(b). Is the Transformation on that Rotates Points about Some line through the Origin

The eigenvalue is and it's eigenspace is all vectors along the line of rotation.