For a polynomial \( p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \) with \( a_i \in \mathbb{F} \) and square matrix \( A \) with entries in \( \mathbb{F} \), define \( p(A) \) by\[p(A) = a_nA^n + a_{n-1}A^{n-1} + \cdots + a_1A + a_0I.\]Let \( A \in M_{n \times n}(\mathbb{C}) \) be diagonalizable and \( c(x) \) the characteristic polynomial of \( A \). Then show (i.e., prove) that \( c(A) = O \), where \( O \) is the \( n \times n \) zero matrix.

Answered: Image /qna-images/answer/7b6c656b-4636-4f4e-ab03-ed26c99b4746.jpg

Answered: For a polynomial p(x) = anx" + an-1xn-1…

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

Let V = P2(R) be the space of polynomials of degree at most 2 with
1 X1 V 1 X2 X3 det V = (x2 – x1)(x3 – X)(x3 – x2) Let x1, x2 and x3 be fixed numbers all distinct. Matrix V can be used to find an interpolating quadratic polynomial for the points (x1, yı), (x2, y2) and (x3, y3), where y , y2 and y3 are arbitrary prove the existence of an interpolating polyno- mial p(t) = co +c;t + c2t² such that p(x1) = y1, p(x2) = V2 and p(x3) = y3 ·
For the following graph, G₁, b a e g
Find the characteristic polynomial of the matrix [58 3.
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.
Consider a linear transformation T from R? to R² for which 7 (C) - [- ma 7 (1) - [- (H)-- T and T Find the matrix A of T. A: %3D
Question : For which real values of a do the polynomials P1(t) = at² - t -; P2(t) = -² + at -; P3(t) = -² - + a form %3D a linearly dependent set in P2?
LetT: P₂ → P3 be the transformation that maps a polynomial p(t) into the polynomial (5t + 2)p(t)) Find the matrix for T relative to the following bases B and C. C={C₁,C₂,C3,C4} = {1,t,t²,t³} B=(b₁,b₂,b3} = {1,t,t²}, 500 250 025 002 a. b. C. d. 5200 05 20 0052 200 520 052 005 2500 0250 0025
Let G be the function from the space of 2 × 2 matrices to itself defined by: a b G (2) - [d 3a+c -b Prove that G is a linear transformation.

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