defined by Let n ≥2. Consider the reversing operator R:R" R(a1, a2,,an-1, an) = (an, an-1,..., a2, a1). → Rn We denote by E₁ and E-1 the eigenspaces of eigenvalues λ = 1 and λ = −1, respec- tively. (1) If n is even, find a basis of E₁ and a basis of E-1, and determine whether or not R is diagonalizable. (2) If n is odd, find a basis of E₁ and a basis of E-1, and determine whether or not R is diagonalizable.

defined by Let n ≥2. Consider the reversing operator R:R" R(a1, a2,,an-1, an) = (an, an-1,..., a2, a1). → Rn We denote by E₁ and E-1 the eigenspaces of eigenvalues λ = 1 and λ = −1, respec- tively. (1) If n is even, find a basis of E₁ and a basis of E-1, and determine whether or not R is diagonalizable. (2) If n is odd, find a basis of E₁ and a basis of E-1, and determine whether or not R is diagonalizable.

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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26. Prove: If {u1, u2, . . . , u,} is an orthonormal basis for R", and if A can be expressed as A = c,u,u + c,u,u + · · + c,,4,U, then A is symmetric and has eigenvalues c1, C2, . . . , Cn .
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10. Suppose that |a) = E"-1 ax|k), |B) = E"-, bu|k). Show that (la)(8|)aj = a,b,, and that this means that the matrix representation in the given orthonormal basis satisfies |a)(3| = a b°, where a= [a a,]", b= [b, ..
6. Let A be an n x n hermitian matrix, i.e. A = A*. Assume that all n eigenvalues are different. Then the normalized eigenvectors { v; : j = 1,...,n} form an orthonormal basis in C". Consider B:= (Ax – ux, Ax – vx) = (Ax – ux)*(Ax – vx) | where (, ) denotes the scalar product in C" and µ,v are real constants with µ 0.
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of Let W be the set of all 2 × 2 matrices [a b] C d basis for W is S = (a) • X (b) following questions: • Y = • r {[ || (c) where, c = 0 and d = 2a - 3b. A || •S= 1 x ] []} " S О У I dim(W) - Answer the