1. (a) Use Jordan form to prove that every complex nx n matrix T is similar...
Complex Jordan Form to Real Jordan Form. Below is the Jordan form for some matrix A. However, it is in complex form. What is the Real Jordan Form of this matrix? That is, a Jordan form with all entries being real values. ( -1) 1 Ο -1 Ο Ο Ο Ο Ο Ο Ο Ο 1 +Ι- Ο Ο 1-I-)
2. For n . define functions T inductivelv such that 0, 1, 2, . . . (cosx) = cos(nx), with Folz) 1. (a) Prove that Tn is a polynomial for every n and compute its degree. b) Prove the recursion formula (c) Compute the integral dr 山 for every n, m E N 2. For n . define functions T inductivelv such that 0, 1, 2, . . . (cosx) = cos(nx), with Folz) 1. (a) Prove that Tn is...
Question 4 [35 marks in totalj An n x n matrix A is called a stochastic matrix if it! satisfies two conditions: (i) all entries of A are non-negative; and (ii) the sum of entries in each column is one. If the (,) entry of A is denoted by any for ij € {1, 2,...,n}, then A is a stochastic matrix when alij 20 for all i and j and I j = 1 for all j. These matrices are...
Prove that trace (A*A)=||A||F2 Definition. Let A be an m x n complex matrix. Its Frobenius noTm, denoted by ||A||F, is defined as follows: m ΣΣΑ Β) / . A|| F = j=1 k=1 1. (7 marks) Let A be as above. Prove that trace(AH A) || A||?
is similar tO its transpose i.e., there exists an invertible matrix Q E Mn (C) so that (b) (5 marks) Deduce that if A є Mn(C), then A is similar to its transpose AT.
Let A be an nx n matrix. Select all of the following that are equivalent to the statement: A is invertible. The homogeneous equation Ax-0 has a nontrivial solution. The echelon form of A has a pivot in every row and every column. The columns of A are linearly dependent For any vector b in R", Ax-b has a unique solution. The linear transformation x Ax is 1-1 and onto. A is nonsingular.
Problem 1. Let A be an m x m matrix. (a) Prove by induction that if A is invertible, then for every n N, An is invertible. (b) Prove that if there exists n N such that An is invertible, then A is invertible. (c) Let Ai, . . . , An be m x m matrices. Prove that if the product Ai … An is an invertible matrix, then Ak is invertible for each 1 < k< n. (d)...
Problem 4. Give an example of a linear operator T on a finite-dimensional vector space such that T is not nilpotent, but zero is the only eigenvalue of T. Characterize all such operators. Problem 5. Let A be an n × n matrix whose characteristic polynomial splits, γ be a cycle of generalized eigenvectors corresponding to an eigenvalue λ, and W be the subspace spanned by γ. Define γ′ to be the ordered set obtained from γ by reversing the...
Let A be the nx n matrix as listed below 1 1 1 1 1 3 3 1 3 6 3 А ... 6 . ... 1 3 6 3(n-1)] (a) Use the appropriate row replacement operations to zero out the first pivot then use the appropriate row replacement operations to zero out the second pivot column. (b) Observe the resulting matrix from (a) is a block matrix of the form: A= X Y 0 Z State your resulting matrix...
Problem 4: Suppose A = (ai)nxn is a symmetric matrix (i.e. the transpose of A agrees with itself) and a11 +0. After we use a11 to eliminate a21, ... , Anl, we obtain a matrix of the following form: (n-1)-matrix. Here c is an (n-1)-dimensional column vector and ct is its transpose, while B is an (n-1) Prove that B is also symmetric.