4.What does it mean for matrices A and B to be similar? Prove that, if A and Bare similar, then they have the same characteristic polynomial.Consider the mapping T : R³ → R³, defined by T(x) = Cx where10 4c-(19)C =324Find the characteristic polynomial of C. The three eigenvectors of Care (1, 1, 1), (−1, 0, 1)and (2, 1,0). With this information, find a matrix P such that P-¹CP is diag-onal, where the diagonal elements are in increasing order.

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Answered: 4. What does it mean for matrices A and…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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For the linear map given by the matrix A below, you may use without calculation that chT (x) = (x − 1)3. Determine (i) the minimal polynomial, (ii) a Jordan basis (iii) the Jordan normal form. 432 A=−4 −3 −3 112
i. Consider the matrix A = The characteristic polynomial of A is p(A) = X² – A–1. What happens when we substitute A into this polynomial? I.e., what is A² – A – I?
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1. For the following quadratic forms (i) write down the symmetric matrix A (so the form is equal to x"Ax), (ii) find matrices P, orthonormal, and D so that D = PT AP is diagonal, (iii) write the quadratic form in terms of y1,..., Yn where x = Py and (iv) decide whether the form is positive/negative definite/semidefinite or indefinite. (a) 2x – 2x1x2 + 2x3 (b) —г? + 4122 — 4л2 (c) 3xỉ + 4x3 + 5x3 + 4x1x2 – 4x2x3 (it may help to show that A = 1 is a root of the characteristic polynomial of A). -
Find the linear transformation T: P) + P1) such that T has eigenvalues -2 and 3 associated with the polynomials p(x) = 3a + az and g(x) = -2b + bx, respectively. (Here a and b are arbitrary real numbers)
Give example or show that this thing doesn't exist a. A 3x3 real matrix with exactly one complex eigenvalues a tbi with b ±0 b. A linear transformation whose domain is R² and whose is the line x +y = 1 Kernel C. A rank 2, diagonalizable, 3 x3 matrix that is not diagonal itself CS Scanned with CamScanner
T: V ---> V is a linear operator with a B and B' ordered bases for V a. Find the matrix representation for T relative to the ordered bases B and B' b. Find T(v), using a direct computation and using the matrix representation
Let A be an n x n matrix over C. Let f be an entire function, i.e. an analytic function on the whole complex plane, for example exp(z), sin(z), cos(z). Using the Cayley-Hamilton theorem we can write ƒ(A) = an−1A”−¹ + an−2A¹−²+...+ a₂Ã² + a₁A+aoIn (1) where the complex numbers ao, a₁, ..., an-1 are determined as follows: Let r(λ) := An−1\”−¹ + an-21²-2+...+ a₂λ² + a₁λ + ao which is the right-hand side of (1) with A¹ replaced by X³, where j = 0, 1, ….., n— 1 of each distinct eigenvalue λ; of the matrix A, we consider the equation ƒ(λj) =r(λj). (2) If X, is an eigenvalue of multiplicity k, for k> 1, then we consider also the following equations ƒ'(A)|a=a; = r′(A)|a=a; ƒ"(A)|a=a; = r″(A)|a=a; ...... f(k-1) (X) = (k-1)(A) (A)|x 1x=x₂ Apply this technique to find exp(A) with с A = = (ε c), C C * |x=x₂ CER, C# 0.
Let T:P2 → P2 be defined by T(a + bx + cx²) = (5a + 6b + 2c) – (b + 8c)x + (a – 2c)x². Find bases for the eigenspaces of T.
2. Prove Cayley-Hamilton Theorem for a diagonalizable matrix, A: Let A be diagonalizable and p(x) be its characteristic polynomial; i.e. value of A. p(A) : = 0 for A, eigen- Show that p(A) = 0.
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.
IV. Let B₁ = {V₁, V2, V3} and B₂ = {w₁, w2, w3} be ordered bases of R³. The change of basis matrix from B₂ to B₁ is [213] P = 0 1 2 [103] 1. Write w₁ w₂ + 2w3 as a linear combination of v₁, U2, U3. 2. Determine P-1 using elementary row operations. 3. Suppose f: R³ R³ is the linear transformation such that f(v1) = 2v1 + v3, f(v2) = v2 +2v3, f(v3) = v₁ + 2v2. Find the matrix representation A off with respect to B₁. 4. What is the matrix representation C of f with respect to B₂?

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