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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Find the smallest integer value for K so that the linear ,3 operator T:R°→R° given by T(xy,z) = (x + y +z,kx +y – z,x + y +z) be degonalizable. Then, for the value of K found, determine a diagonal matrix similar to the matrix of the operator T on the canotic basis of the R
Consider the function T : P2 (R) → P2(R) given by T(a + bx + cx²) = (a – 6+ c) + bx + (-6 – 2c)a². Let T be a linear transformation and eigenvalues of T are A = 1 and A= 2. Find a basis for E2(T) and the geometric multiplicity of A = 2.
Let T: V → V be a linear operator whose characteristic polynomial is pT(x)=x2(x−1)3(x+2)2. Judge each item below as true or false. (A) If the minimal polynomial of T is pT(x) then [T]BB is diagonal for some basis B of V.(B) Let A be the matrix of T with respect to some basis of V. Then, A2(A−I)3(A+2I)2 ≠ 0.
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.
6. Find a basis of the null space of each of the differential operators below, and verify that the dimension of the null space is equal to the degree of the operator. (Use complex numbers where necessary for simplicity.) (a) D² − 4D + 3 (b) D² + 2D + 2 (c) D³ (d) Dn (e) Dª + 4.
Let d : P5(R) → P5(R) denotes the first derivative. Show that d is a linear map. Choose a basis for P5(R) and find the matrix M(d). Use the matrix M(d) to find the derivative of f(x) = −9x^5 + 11x^4 −4x^2 +5x−1.
Suppose S: R² → R² and T: R² → R² are linear transformations given by 3x y (0)-[-1-(C)-[22] T -x-3y] (a) Find the standard matrix for S (assuming the stand basis for R³ and for R²). (b) Find the standard matrix for T (assuming the standard basis for R² and for R³). (c) Show that S and T are invertible. (d) Show that T is the inverse of S. (e) What is the standard matrix for T(S(z)) = To S(1), where I = R²?
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Let A be the matrix defined by [4 1 A = 0 -3 Let T : R? → R² be the map defined by T(x) = Ax. (2, –3). Compute A10(x) by using the matrix (b) M(T) above. Don't forget to convert x to the basis B, and then Let x = convert back.
[1 4 and find its inverse. Also express 2 3 A´ – 4A* Verify Cayley-Hamilton theorem for the matrix A = - 7A' + 11A? - A – 101 as a linear polynomial in A
[*:* ] a + 2 a (10) Find all a E R for which the matrix is singular. 1 a

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