Eigenvalues & Eigenvectors

**Order Instructions:**

Hi Admin,

Please send me the answers by 28th June.

Thanks,

Customer

**SAMPLE ANSWER**

**Eigenvalues & Eigenvectors**

*Problem 1*

The matrix has eigenvalues1=3 and 2=−2. Let’s find the eigenvectors corresponding to 1=3. Let v=*v*2*v*1. Then(*A*−3*I*)v=0 gives us

2−3−1−4−1−3*v*1*v*2=00 from which we obtain the duplicate equations

−*v*1−4*v*2−*v*1−4*v*2=0

If we let *v*2=*t*, then*v*1=−4*t*. All eigenvectors corresponding to1=3are multiples of1−4 and thus the eigenspace corresponding to1=3is given by the span of1−4. That is,1−4is a basis of the eigenspace corresponding to 1=3.

Repeating this process with 2=−2, we find that

4*v*1−4*V*2−*v*1+*v*2=0

If we let *v*2=*t *then *v*1=*t *as well. Thus, an eigenvector corresponding to2=−2 is 11 and the eigenspace corresponding to 2=−2 is given by the span of11. 11is a basis for the eigenspace corresponding to 2=−2.

*Problem 3*

*Problem 5*

*Problem 14*

*Problem 22*

*Problem 25*

Matrices A_{2 }and A_{3} cannot be diagonized because for a square matrix A, wherever A is similar to diagonal matrix then the matrix is diagonizable.

**Problem 15-24: Eigenvalues & Eigenvectors Matrices **

*Problem 15*

Since all entries are ≥ 0 and each column sums to 1, this A is a Markov matrix. Thus we know that λ1 = 1 is an eigenvalue. Since tr(A) = λ1 + λ2 = 3/2, we conclude λ2 = 1/2 is another eigenvalue. We diagonalize it using the matrix S of eigenvectors:

= → =

This last matrix product equals

*Problem 19*

- False
- True
- True
- False

**References**

Bhatti, M. A. (2012). *Practical Optimization Methods with Mathematica Applications. *New York, NY: Springer.

Edwards, C. H., & David E. Penney, D. E. (2009). *Differential Equations: Computing and Modeling. *Upper Saddle River, CA: Pearson Education, Inc.

Shores, T. S. (2007). *Applied linear algebra and matrix analysis. *New York, NY: Springer Science+Business Media, LLC.

Strang, G. (2003). *Introduction to linear algebra. *Wellesley, MA: Wellesley-Cambridge Press

Strang, G. (2006). *Linear algebra and its applications. *Belmont, CA: Thomson, Brooks/Cole Publishers.

Strang, G. (2009). *Eigenvalues and Eigenvectors. *Boston, MA: Lord Foundation of Massachusetts.

Zhang, F. (2009). *Linear Algebra: Challenging Problems for Students*, Baltimore, MA: The Johns Hopkins University Press.http://nsuworks.nova.edu/cnso_math_facbooks/3/

We can write this or a similar paper for you! Simply fill the order form!