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.
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.
An eigenvalue is and it's eigenspace is all vectors normal to the line that points are reflected across which start at the origin.
The eigenvalue is and it's eigenspace is all vectors along the line of rotation.
Problem 23
Find a basis for the eigenspace corresponding to the given eigenvalue.
Problem 24
Mark each of the following statements as either true or false.
- [ ] If for some scalar , then is an eigenvector of .
- [ ] If and are linearly independent eigenvectors, then they correspond to distinct eigenvalues.
- [ ] The eigenvalues of a matrix are on its main diagonal.
- [x] An eigenspace of is a null space of a certain matrix.
Problem 25
It can be shown that the algebraic multiplicity of an eigenvalue is always greater than or equal to the dimension of the eigenspace corresponding to . Find in the matrix below such that the eigenspace for is two-dimensional: