3. Compute a spectral decomposition for the following matrix D e Mat3xa(R) as it has been doneduring the lecture, i.e., find an invertible matrix C such that C-'DC is a block matrix where everyblock corresponds to an irreducible factor of the minimal polynomial.6 -1 -4D:=| 11 -2-6-5

Answered: Image /qna-images/answer/ec76e335-e7fb-43d7-be2a-b49ec5c5c089.jpg

Answered: 3. Compute a spectral decomposition for…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

Q2. Consider a state space system with 1 c = [1 2 0]. A = ㅇ 1 B = -3 5 For each of the following desired polynomials, find a gain matrix K such that the characteristic polynomial of A – BK is same as the desired characteristic polynomial. а. s3 + 3s2 – 5s +2 b. s3 + s? +s +1 c. s + 2s2 + 3s +4
Help with number 4 please, a and b
Suppose that T: R" → Rm is defined by T() = A for each of the following matrices below: [40] E = 0 0 02 0 D = 0 -1 0 0 0 0.5 [201] F = 10 3 01 (a) Rewrite T: R" → Rm with correct numbers for m and n for each transformation. What is the domain and codomain of each transformation? (b) Find some way to explain in words and/or graphically what each transformation does as it takes vectors from R to Rm. You might find it t helpful to try out a few input vectors and see what their image is under the transformation. This might be difficult, but an honest effort will give you credit. (c) For the transformation, can you get any output vector? (Any vector in R™) i. If so, explain why you can get any vector in Rm. ii. If not, give an example of an output vector you can't get with the transformation and explain why.
(2) Find the examples that satisfy the statement in the given items and show some work how they are correct examples (a) A 4 x 4 matrices A that is not digonalizable. (b) (c) Two matrices A and B that have the same characteristic polynomial but are not similar. Two diagonal matrices D₁ and D₂ that are similar but they are not equal. (d) Two similar matrices A and B such that A is orthogonally digonalizable but B is not orthogonally diagonalizable.
Q 1. Find an SVD of the matrix 7 0007 A 0 3 00 0000 Q 2. Use the SVD of A to prove that ||A| = trace (ATA). Q3. Prove that A+ A is an orthogonal projection where A+ is the pseudo-inverse of A. Q4. Use SVD of A to show that if matrix A is full row-rank then the pseudo-inverse is given by A+ = AT (AA)-¹. Verify that this happens to be the right inverse of the matrix A.
2. Let A be an nxn matrix over R. Consider the LU-decomposition A = LU, where L is a unit lower triangular matrix and U is an upper triangular matrix. The LDU-decomposition is defined as A = LDU, where L is unit lower triangular, D is diagonal and U is unit upper triangular. Let 2 4 -2 A = 4 9 -3 -2 -3 7 Find the LDU-decomposition via the LU-decomposition.
Suppose that B = S-¹AS for some n x n matrices A, B and some n x n invertible matrix S. a. Show that if is in ker B, then Sa is in ker A. b. Show that the linear transformation T (*) = S from ker B to ker A is an isomorphism. c. Show that nullity A = nullity B and rank A = rank B.
Let A be a 2 x 2 matrix with characteristic polynomial c₁(x) = (x − 5)². If A is diagonalizable, what must be the RREF of 5I - A? 16.9
2. Find the matrix A of T so that T ([]) = A[*].
Let T: R² R² be defined by T and C Given Pc 3 -{]-[H]} -1223) , use the Fundamental Theorem of Matrix Representations to find [T](PB(u)). [T](PB (u)) 7 [4₁]) B-{B] R]} = Let u = = Ex: 5 [-3x1 + 2x₂ / -4
Q) Answer the following. (50 Marks) (a) Determine the inverse of the matris - A- S 3 51 b) Find the Transpose of a matrix - A-(2 5 1 A- i3 -24 24 Q2) Answer the folloning --(50 Marko) 1- limr'n 2- lim 3y+2 - lim lim 4-

Recommended textbooks for you

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Mathematics For Machine Technology

Advanced Math

ISBN:

9781337798310

Author:

Peterson, John.

Publisher:

Cengage Learning,

Basic Technical Mathematics

Advanced Math

ISBN:

9780134437705

Author:

Washington

Publisher:

PEARSON

Topology

Advanced Math

ISBN:

9780134689517

Author:

Munkres, James R.

Publisher:

Pearson,

SEE MORE TEXTBOOKS

3. Compute a spectral decomposition for the following matrix D E Matax3(R) as it has been done during the lecture, i.c., find an invertible matrix C such that C-DC is a block matrix where every block corresponds to an irreducible factor of the minimal polynomial. 6 -1 -4 D:=| 11 -2 -6 -1 -5