**Assignment 3***• *All regular assignments are due in the class on the specified date.*• *For assignment(s) involving coding,**– **Must be done with ** Matlab**. Each must include a summary report.

**–**

**– **Name your code(s) as follows: Add Progx and your last name (or first 4 letters if too long) to the beginning of each file name. E.g., if the 3rd problem is a programming assignment, then my file names will be something like: * P rog*3

*3*

*Chen main.m, P rog**.*

*Chen newtion.m***– **Email your codes with instructions to the grader before 11:30PM of the due date ** If not specified otherwise, all the problems are from the textbook.**1. Read Chapter 4, 52. Determine the following is true or false, and justify your answer(a)

If * Ax *=

*and*

*λx**is nonsingular, then*

*A*__1__is an eigenvalue of

*A*^{−}^{1}(b) If

*is nonsingular, then for any induced norm,*

*A*

*"A*^{−}^{1}

*=*

*"**13. Which of the following matrices are necessarily orthogonal? Justify your answer(a) Permutation matrices, which are obtained by permuting rows or columns of the identity matrix, so that in each row and each column we still have exactly one value equal to 1;(b) Symmetric positive definite matrices;(c) Nonsingular matrices;(d)*

*"A"−*Diagonal matrices.4. (a) Under what condition, the matrix * c s *is orthogonal?

*(b) Given rotation of the form.*

*−s c**Σ .*

*c s**1Σ = .*

*α**Σ*

*α*

*.**−s c α*_{2} 0

That is, given a vector we rotate it (while preserving its length) so that one of its component is zeroed out. Express * α *in terms of

*α*_{1}

*, α*_{2}, and find the values of

*.*

*c, s*5. Show that the eigenvalues of an orthogonal matrix are equal to 1 in modulus, and find the eigenvalues of a projector * P *. (

*A projector is a square matrix that satisfies: P*^{2}=

*)*

*P*6. Page 134: #1

7. Let * A, T *be two nonsingular

*real matrices, and*

*n × n** T A *=

*L U*where * L *is unit lower triangular, and

*is upper triangular. Write an algorithm that will solve*

*U** A x *=

*b*in * O*(

*n*^{2}) complexity.

8. Page 138: #17

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