• Matrix N € Cnxn is said to be nilpotent whenever Nk = Onxn for some positive integer k. • The smallest positive integer k such that Nk = Onxn, denoted by index (N) = k, is referred to as the index of nilpotency. 1. Let N be an n x n matrix. Prove that the following statements are equivalent. (a) N is nilpotent. (b) The only eigenvalue of N is λ = 0. (c) The characteristic polynomial of N is X". (d) The minimal polynomial of N is №k for some integer k ≤ n.

Q: Consider the following. A =    18 −80   4 −18,   P =    −4 −5   −1 −1 (a) Verify that A is…

A:

Q: Kailashben

A: Step 1: Step 2: Step 3: Step 4:

Q: Consider an N x N matrix A  with N orthonormal eigenvectors xi such that Axi = λi xi  , where the λi…

A:

Q: For which value of k does the matrix have one real eigenvalue of multiplicity 2? k = A = 6 -8 2

A:

Q: Evaluate the expression a+b a+b when a=23a=23 and b=45b=45. Write in simplest form

A: # the expressions are printed twice , taking them once given a=23 and b=45 The value of a+b is as…

Q: Consider the nxn Hilbert matrix defined by 1 H = 1 13 1 n n+1 1 3 1 4 1-n 1 n+1 n+2 1 n+2 1 2n-1…

A: The answer is hij=1i+j-1

Q: (1) If A is diagonalizable then cA is diagonalizable for any constant c. (i1) If the characteristic…

A:

Q: Suppose that A is an n x n matrix such that Ak is the n x n zero matrix for some positive integer k.…

A:

Q: Which of the following statements is(are) true if

A: Introduction: Only zero and one are the eigenvalues of the identity matrix and the zero matrices,…

Q: The matrix A=[7    k]      [-8 −4]   has two distinct real eigenvalues if and only if k<_____ (Find…

A:

Q: Let A = (a) be a n x n matrix where ai; is the entry of the matrix on row i and column j. It is…

A:

Q: Let A and B be an n x n matrices. (a) (b) (c) BA. State the definition of an eigenvector of an n x n…

A:

Q: 3. An n x n matrix A is called nilpotent if there is some k ≥ 0 such that Ak = 0 (i.e., Ak = (n x n)…

A:

Q: 8. Suppose 2 is an eigenvalue of an invertible matrix A. Prove that is an eigenvalue of kA-¹ for…

A:

Q: Q1.10 Assume that you know matrix Z is a 2x2 real matrix. the matrix is known to have the eigenvalue…

A:

Q: Suppose A is a 2 x 2 real matrix that has two eigenvalues A₁ and ₂, and X₁ is not equal to X₂.…

A:

Q: 2. Without using Sage, find • the characteristic polynomial the eigenvalues (include the…

A: Since you have asked multiple questions, we can solve first question for you. If you want other…

Q: 7.- Diagonalize the following matrix, defining the characteristic polynomial, the eigenvalues, the…

A: