See image for wuestion LetA =-1(a) Find the characteristic polynomial and the eigenvalues of A over the complex numbers.(b) Find a basis of the eigenspace of one of the eigenvalues of A over the complex numbers.(c) Without any more computation, determine a basis for the eigenspace of the other eigenvalue ofA over the complex numbers. Explain briefly how you determined your answer.(d) Find a matrix P and a matrix D such that P-¹AP=D. [Hint: you do not have to computeP¹. Note that you can check your work by checking AP = PD.]

Answered: Image /qna-images/answer/aec5c3ce-97b1-4b97-a1de-ba9d80c7cd5f.jpg

Answered: Let A = -1 (a) Find the characteristic…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.1: Introduction To Eigenvalues And Eigenvectors

Problem 38EQ

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Please help, in order to review for a test:
show that(2+i, 1+i) is not a complex scalar multiple of (3+2i,4+i)
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Find all complex roots of the equation z6+z5+z4+z3+z2+z+1=0.
Given a matrix where α and β are non-zero complex numbers, find its eigenvalues and eigenvec- tors. Find the respective conditions for (a) the eigenvalues to be real and (b) the eigenvectors to be orthogonal. Show that the conditions are jointly satisfied if and only if A is Hermitian.
Complex numbers have a representation as two by two matrices with real entries. In this problem you will be asked to show that the matrix operation indeed reproduces the complex number operations. For any complex number z assign a matrix Mz as follows: z=a+ib→Mz= a -b ; b a. (a) Show that Mz1 + Mz2 = Mz1+z2 and cMz = Mcz, for any complex numbers z, z1, z2 and real number c. (b) Show that Mz1xMz2 = Mz1z2 , for any complex numbers z1, z2. c) Show that M1/z = Mz^−1, for any nonzero complex number z. (d) Find an expression for det(Mz) and Mz^T in terms of operations of complex numbers. (e) Find the eigenvalues and corresponding eigenvectors of Mz for z = a + ib. (Note that you can only use the techniques used in the section 4.1). I HAVE ATTACHED IMAGES FOR BETTER UNDERSTANDING
[EX5 Q1 Diagonalisation] could u please answer the circled question step by step on image1? And the solution is on the image2, thx :)
Find all the eigenvalues (real and complex) of the matrix The eigenvalues are M = -4 -5 5 6 -1 -2 2 . (Enter your answers as a comma separated list.)
Suppose the product z1z2 of two complex numbers is a nonzero real constant. Show that z2 = kž1, where k is a real number.

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