Linear Algebra/Eigenvalues and Eigenvectors/Solutions
Solutions[edit]
 Problem 1
For each, find the characteristic polynomial and the eigenvalues.
 Answer
 This
 ; ,
 ; ,
 ;
 ;
 This exercise is recommended for all readers.
 Problem 2
For each matrix, find the characteristic equation, and the eigenvalues and associated eigenvectors.
 Answer
 The characteristic equation is . Its roots, the eigenvalues, are and . For the eigenvectors we consider this equation.
 The characteristic equation is
 Problem 3
Find the characteristic equation, and the eigenvalues and associated eigenvectors for this matrix. Hint. The eigenvalues are complex.
 Answer
The characteristic equation
has the complex roots and . This system
For Gauss' method gives this reduction.
(For the calculation in the lower right get a common denominator
to see that it gives a equation.) These are the resulting eigenspace and eigenvector.
For the system
leads to this.
 Problem 4
Find the characteristic polynomial, the eigenvalues, and the associated eigenvectors of this matrix.
 Answer
The characteristic equation is
and so the eigenvalues are (this is a repeated root of the equation) and . For the rest, consider this system.
When then the solution set is this eigenspace.
When then the solution set is this eigenspace.
So these are eigenvectors associated with and .
 This exercise is recommended for all readers.
 Problem 5
For each matrix, find the characteristic equation, and the eigenvalues and associated eigenvectors.
 Answer
 The characteristic equation is
 The characteristic equation is

 This exercise is recommended for all readers.
 Problem 6
Let be
Find its eigenvalues and the associated eigenvectors.
 Answer
With respect to the natural basis the matrix representation is this.
Thus the characteristic equation
is . To find the associated eigenvectors, consider this system.
Plugging in gives
 Problem 7
Find the eigenvalues and eigenvectors of this map .
 Answer
, ,
 This exercise is recommended for all readers.
 Problem 8
Find the eigenvalues and associated eigenvectors of the differentiation operator .
 Answer
Fix the natural basis . The map's action is , , , and and its representation is easy to compute.
We find the eigenvalues with this computation.
Thus the map has the single eigenvalue . To find the associated eigenvectors, we solve
to get this eigenspace.
 Problem 9
 Prove that
the eigenvalues of a triangular matrix (upper or lower triangular) are the entries on the diagonal.
 Answer
The determinant of the triangular matrix is the product down the diagonal, and so it factors into the product of the terms .
 This exercise is recommended for all readers.
 Problem 10
Find the formula for the characteristic polynomial of a matrix.
 Answer
Just expand the determinant of .
 Problem 11
Prove that the characteristic polynomial of a transformation is welldefined.
 Answer
Any two representations of that transformation are similar, and similar matrices have the same characteristic polynomial.
 This exercise is recommended for all readers.
 Problem 12
 Can any non vector in any nontrivial vector space be a eigenvector? That is, given a from a nontrivial , is there a transformation and a scalar such that ?
 Given a scalar , can any non vector in any nontrivial vector space be an eigenvector associated with the eigenvalue ?
 Answer
 Yes, use and the identity map.
 Yes, use the transformation that multiplies by .
 This exercise is recommended for all readers.
 Problem 13
Suppose that and . Prove that the eigenvectors of associated with are the non vectors in the kernel of the map represented (with respect to the same bases) by .
 Answer
If then under the map .
 Problem 14
Prove that if are all integers and then
has integral eigenvalues, namely and .
 Answer
The characteristic equation
simplifies to . Checking that the values and satisfy the equation (under the condition) is routine.
 This exercise is recommended for all readers.
 Problem 15
Prove that if is nonsingular and has eigenvalues then has eigenvalues . Is the converse true?
 Answer
Consider an eigenspace . Any is the image of some (namely, ). Thus, on (which is a nontrivial subspace) the action of is , and so is an eigenvalue of .
 This exercise is recommended for all readers.
 Problem 16
Suppose that is and are scalars.
 Prove that if has the eigenvalue with an associated eigenvector then is an eigenvector of associated with eigenvalue .
 Prove that if is diagonalizable then so is .
 Answer
 We have .
 Suppose that is diagonal. Then is also diagonal.
 This exercise is recommended for all readers.
 Problem 17
Show that is an eigenvalue of if and only if the map represented by is not an isomorphism.
 Answer
The scalar is an eigenvalue if and only if the transformation is singular. A transformation is singular if and only if it is not an isomorphism (that is, a transformation is an isomorphism if and only if it is nonsingular).
 Problem 18
 Show that if is an eigenvalue of then is an eigenvalue of .
 What is wrong with this proof generalizing that? "If is an eigenvalue of and is an eigenvalue for , then is an eigenvalue for , for, if and then "?
 Answer
 Where the eigenvalue is associated with the eigenvector then . (The full details can be put in by doing induction on .)
 The eigenvector associated wih might not be an eigenvector associated with .
 Problem 19
Do matrixequivalent matrices have the same eigenvalues?
 Answer
No. These are two samesized, equal rank, matrices with different eigenvalues.
 Problem 20
Show that a square matrix with real entries and an odd number of rows has at least one real eigenvalue.
 Answer
The characteristic polynomial has an odd power and so has at least one real root.
 Problem 21
Diagonalize.
 Answer
The characteristic polynomial has distinct roots , , and . Thus the matrix can be diagonalized into this form.
 Problem 22
Suppose that is a nonsingular matrix. Show that the similarity transformation map sending is an isomorphism.
 Answer
We must show that it is onetoone and onto, and that it respects the operations of matrix addition and scalar multiplication.
To show that it is onetoone, suppose that , that is, suppose that , and note that multiplying both sides on the left by and on the right by gives that . To show that it is onto, consider and observe that .
The map preserves matrix addition since follows from properties of matrix multiplication and addition that we have seen. Scalar multiplication is similar: .
 ? Problem 23
Show that if is an square matrix and each row (column) sums to then is a characteristic root of . (Morrison 1967)
 Answer
This is how the answer was given in the cited source.
If the argument of the characteristic function of is set equal to , adding the first rows (columns) to the th row (column) yields a determinant whose th row (column) is zero. Thus is a characteristic root of .
References[edit]
 Morrison, Clarence C. (proposer) (1967), "Quickie", Mathematics Magazine 40 (4): 232.
 Strang, Gilbert (1980), Linear Algebra and its Applications (Second ed.), Harcourt Brace Jovanovich.