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1. Suppose U is a unitary matrix with eigenvalues Ai a) Pro...

1. Suppose U is a unitary matrix with eigenvalues Ai a) Prove that = 1. b) Prove that |det(U)| = 1. 2. to be idempotent if A2 = A. Prove that if a matrix A is idempotent then rank (A -1) = mullity(A). Note: A matrix A is said 3. Suppose that A is an m x n matrix and let min (m,1 n) denote the ...

3.1 [Calculus and optimization For a given vector b € R...

3.1 [Calculus and optimization For a given vector b € Rd, Consider the function Fb : (Rdxd x Rd R defined as: Fb: (A,v) I UTAU+656. (i) What is OFb/8A? (ii) What is OFb/Ov? (iii) For a fixed invertible and symmetric matrix A, what value of U minimizes the function Fb? (iv) For a fixed i...

Is the Cholesky factorization of the matrix Find 2 - 1 0 ...

Is the Cholesky factorization of the matrix Find 2 - 1 0 0 = -1 0 - 2 1 - 2 1 -1 0 3 T4 - 1 2 Problem 1. Let be the tridiagonal matrix of the form 1 -1 1 2 -1 -1 - 2 -1 A = -1 2 -1 - 1 2 (a) Show that A has a Cholesky decomposition A = LLT with L of the form 1 a 1 L = a 1 a ...

Problem 3. Consider the matrix A= - 0 1 1 -3 -3 3 1 1 2 1 0...

Problem 3. Consider the matrix A= - 0 1 1 -3 -3 3 1 1 2 1 0 0 1 0 1 2 0 1 (a) Compute the singular value decomposition of A. (Hint: Notice that A = BCT with BT B and CT C both diagonal.) (b) Find the best rank-one approximation to A in the operator 2-norm. 2. (10 pts) Let A be the product A 0...

Problem 1: Determine whether the following matrix is orthogo...

Problem 1: Determine whether the following matrix is orthogonal: A= N/A Street, 1/6 Problem 2: (a) Prove that the transpose of an orthogonal matrix is also orthogonal. (b) Explain why the rows of an n x n orthogonal matrix also orthogonal. Problem 3: Show that if AT = - A is any skew-symmetri...

The purpose of this project is to demonstrate how concepts f...

The purpose of this project is to demonstrate how concepts from linear algebra can be applied. Follow the steps below. An eccentric billionaire Bill tells his niece Nancy that he wishes to start distributing some of his wealth to his relatives before he dies in order to avoid inheritance taxes. B...

Problem 1: Determine the eigenvalues and the eigenspaces of ...

Problem 1: Determine the eigenvalues and the eigenspaces of the following two matrices 0 - -1 2 1 and , 1 0 - 1 4 and if one of them is not diagonalizable give a Jordan basis for it. (For the definition of a Jordan basis see my notes Part 3, page 3.) 3 2 0 Problem 2: Let A= 0 1 - 1 . 1 ...

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