**Exercise 5.1.1 ***Find the eigenvalue and eigenvectors of the matrix .Verify that the trace equals the sum of the eigenvalues,and the determinant equals their product. *

**Solution:** Suppose that the eigenvalue is ,then

so or .Eigenvalue correspond to the eigenvector ,eigenvalue correspond to the eigenvector .The sum of the eigenvalue is ,the trace of the determinant is also .The product of the eigenvalue is ,and the determinant of is .

**Exercise 5.1.2** *With the same matrix ,solve the differential equation ,What are the two pure exponential solutions? *

**Solution:** Let .Then the differential equation becomes

Now we find pure exponential solutions.Let ,,then the differential equation becomes

simplify,we get

When ,;When , So the two pure exponential solutions are and The general solution is

**Exercise 5.1.3** *If we shift to ,what are the eigenvalues and eigenvectors and how are they related to those of ? *

**Solution:** The characteristic equation of matrix is .And the characteristic equation of matrix is .So the eigenvalue of matrix is the corresponding eigenvalue of matrix minus ,which are and .

The eigenvectors of two matrices are same.

**Exercise 5.1.4** *Solve ,when is a projection: *

Part of increases exponentially while the nullspace part stays fixed.

**Solution:** Let ,then the differential equation becomes

Now we try to find the pure exponential solution.Suppose that then from the differential equation we know that

simplify it,we get

When ,we could let When ,we could let So the general solution is

**Exercise 5.1.6** *Give an example to show that the eigenvalues can be changed when a multiple of one row is subtracted from another. Why is a zero eigenvalue not changed by the steps of elimination? *

**Solution:** Example:,subtract the second row from the first row,we get .The eigenvalue of are and .The eigenvalue of are and .

A zero eigenvalue is not changed by the steps of elimination because,suppose the steps of elimination turn matrix into matrix ,and ,where is an invertible matrix.,so .So is also an eigenvalue of .

**Exercise 5.1.7** *Suppose that is an eigenvalue of ,and is its eigenvector:. *

- Show that this same is an eigenvector of ,and find the eigenvalue.This should confirm Exercise 3.
- Assuming ,show that is also an eigenvector of ,and find the eigenvalue.

**Solution:** We solve the second part.,so ,so .The eigenvalue of is .

**Exercise 5.1.8** *Show that the determinant equals the product of the eigenvalues by imagining that the characteristic polynomial is factored into *

and making a clever choise of .

**Solution:** In the above formula,let .

**Exercise 5.1.9** *Show that the trace equals the sum of the eigenvalues,in two steps.First,find the coefficient of on the right side of equation (16).Next,find all the terms in *

that involve .They all come from the main diagonal!Find that coefficient of and compare.

**Exercise 5.1.10**

- Construct 2 by 2 matrices such that the eigenvalue of AB are not the products of the eigenvalues of A and B,and the eigenvalues of A+B are not the sums of the individual eigenvalues.
- Verify,however,that the sum of the eigenvalues of equals the sum of all the individual eigenvalues of and ,and similarly for products.Why is this true?

**Solution:**

**Exercise 5.1.12** *Find the eigenvalues and eigenvectors of *

**Solution:**

- The eigenvalues of are and .The corresponding eigenvectors are and .
- The eigenvalues of are and .The corresponding eigenvectors are and .

**Exercise 5.1.13** *If has eigenvalues , has eigenvalues ,and has eigenvalues ,what are the eigenvalues of the by matrix ? *

**Solution:** .This can be figured out by using determinant.

**Exercise 5.1.14** *Find the rank and all four eigenvalues for both the matrix of ones and the checker board matrix: *

Which eigenvectors correspond to nonzero eigenvalues?

**Solution:** ,.The characteristic equation of matrix is .So the eigenvalues of matrix are .

The characteristic equation of matrix is .So the eigenvalues of matrix are .

**Exercise 5.1.15** *What are the rank and eigenvalues when and in the previous exercise are by ?Remember that the eigenvalue is repeated times. *

**Solution:** .One eigenvector of matrix is ,the corresponding eigenvalue of this eigenvector is .So the eigenvalues of are (There are zeros).

.One eigenvector of matrix is another eigenvector of is .The corresponding eigenvalues of these two vectors are and respectively.So the eigenvalues of are ,,(There are zeros).

**Exercise 5.1.15** *What are the rank and eigenvalues when and in the previous exercise are by ?Remember that the eigenvalue is repeated times. *

**Solution:** .One eigenvector of matrix is ,the corresponding eigenvalue of this eigenvector is .So the eigenvalues of are (There are zeros).

.One eigenvector of matrix is another eigenvector of is .The corresponding eigenvalues of these two vectors are and respectively.So the eigenvalues of are ,,(There are zeros).

**Exercise 5.1.17** *Choose the third row of the “companion matrix” *

so that its characteristic polynomial is .

**Solution:**

**Exercise 5.1.18** *Suppose has eigenvalues with independent eigenvectors . *

- Give a basis for the nullspace and a basis for the column space.
- Find a particular solution to .Find all solutions.
- Show that has no solution.(If it had a solution,then would be in the column space.)

**Solution:**

- A basis for the nullspace is .A basis for the column space is .
- A particular solution is .All solutions are of the form ,where is an arbitrary real number.
- Otherwise, would be in the column space,this is an contradiction to the fact that are linearly independent.

**Exercise 5.1.19** *The powers of this matrix approaches a limit as : *

The matrix is halfway between and .Explain why from the eigenvalues and eigenvectors of these three matrices.

**Solution:** The eigenvalues of are and .The corresponding eigenvectors are and

So the eigenvalues of are and .The corresponding eigenvectors are and .

And the eigenvalues of are and ,the corresponding eigenvectors are and .

So .

**Exercise 5.1.25** *From the unit vector ,construct the rank-1 projection matrix . *

- Show that .Then is an eigenvector with .
- If is perpendicular to show that zero vector.Then .
- Find three independent eigenvectors of all with eigenvalue .

**Solution:**

**Exercise 5.1.26** *Solve by the quadratic formula,to reach : *

Find the eigenvectors of by solving .Use .

**Solution:**

,so .The corresponding eigenvectors are and .

**Exercise 5.1.27** *Every permutation matrix leaves unchanged.Then .Find two more ‘s for these permutations: *

**Solution:**

**Exercise 5.1.28** *If has and ,then .Find three matrices that have trace ,determinant ,and . *

**Solution:** All the matrices of the form have trace ,determinant ,and characteristic .

**Exercise 5.1.29** *A by matrix is known to have eigenvalue .This information is enough to find three of these: *

- the rank of ,
- the determinant of ,
- the eigenvalues of ,and
- the eigenvalues of

**Solution:**

- .
- .
- the eigenvalues of are .So the eigenvalues of are .

**Exercise 5.1.30** *Choose the second row of so that has eigenvalues and . *

**Solution:**

**Exercise 5.1.31** *Choose ,so that .Then the eigenvalues are : *

**Solution:**

**Exercise 5.1.32** *Construct any by Markov matrix :positive entries down each column add to .If ,verify that .By Problem 11, is also an eigenvalue of .Challenge:A by singular Markov matrix with trace has eigenvalues . *

**Exercise 5.1.34** *This matrix is singular with rank .Find three ‘s and three eigenvectors: *

**Solution:** ,the corresponding eigenvector is .The rest two eigenvalues are and .

**Exercise 5.1.37** *When ,show that is an eigenvector and find both eigenvalues: *

**Solution:** The eigenvalues are and .

**Exercise 5.1.38** *If we exchange rows and and columns and ,the eigenvalues don’t change.Find eigenvalues of and for .Rank one gives . *

**Solution:**

The eigenvector of corresponding to eigenvalue is .

The eigenvector of corresponding to eigenvalue is .

**Exercise 5.1.39** *Challenge problem:Is there a real by matris(other than )with ?Its eigenvalues must satisfy .They can be and .What trace and determinant would this give?Construct . *

**Solution:**

**Exercise 5.1.40** *There are six by permutation matrices .What numbers can be the determinants of ?What numbers can be pivots?What numbers can be the trace of ?What four numbers can be eigenvalues of ? *

**Solution:** All the six by permutation matrices are

and can be the determinant of .The pivots are .The eigenvalues of can be .

## 近期评论