Let A be a real n × n matrix, and suppose that every leading principal submatrix...

Let A be a real n × n matrix, and suppose that every leading principal submatrix ofA of order k < n is nonsingular. Show that A has an LU-factorisation.

Expert Solution

Colby Messinger answered 2 years ago

Let A ∈ Mat n×n(R) be a real square matrix. (a) Suppose that A is symmetric,...

Let A ∈ Mat n×n(R) be a real square matrix. (a) Suppose that A is symmetric, positive semi-definite, and orthogonal. Prove that A is the identity matrix. (b) Suppose that A satisfies A = −A^T . Prove that if λ ∈ C is an eigenvalue of A, then λ¯ = −λ. From now on, we assume that A is idempotent, i.e. A^2 = A. (c) Prove that if λ is an eigenvalue of A, then λ is equal to 0...

Let A be an n × n real symmetric matrix with its row and column sums...

Let A be an n × n real symmetric matrix with its row and column sums both equal to 0. Let λ1, . . . , λn be the eigenvalues of A, with λn = 0, and with corresponding eigenvectors v1,...,vn (these exist because A is real symmetric). Note that vn = (1, . . . , 1). Let A[i] be the result of deleting the ith row and column. Prove that detA[i] = (λ1···λn-1)/n. Thus, the number of spanning...

Q. Let A be a real n×n matrix. (a) Show that A =0 if AA^T =0....

Q. Let A be a real n×n matrix. (a) Show that A =0 if AA^T =0. (b) Show that A is symmetric if and only if A^2= AA^T

Let A be a diagonalizable n × n matrix and let P be an invertible n...

Let A be a diagonalizable n × n matrix and let P be an invertible n × n matrix such that B = P−1AP is the diagonal form of A. Prove that Ak = PBkP−1, where k is a positive integer. Use the result above to find the indicated power of A. A = 6 0 −4 7 −1 −4 6 0 −4 , A5 A5 =

Let A be a diagonalizable n × n matrix and let P be an invertible n...

Let A be a diagonalizable n × n matrix and let P be an invertible n × n matrix such that B = P−1AP is the diagonal form of A. Prove that Ak = PBkP−1, where k is a positive integer. Use the result above to find A5 A = 4 0 −4 5 −1 −4 6 0 −6

Suppose C is a m × n matrix and A is a n × m matrix....

Suppose C is a m × n matrix and A is a n × m matrix. Assume CA = Im (Im is the m × m identity matrix). Consider the n × m system Ax = b. 1. Show that if this system is consistent then the solution is unique. 2. If C = [0 ?5 1 3 0 ?1] and A = [2 ?3 1 ?2 6 10] ,, find x (if it exists) when (a) b =[1...

Question

Let A be a real n × n matrix, and suppose that every leading principal submatrix...

Solutions

Expert Solution

Related Solutions

Let A ∈ Mat n×n(R) be a real square matrix. (a) Suppose that A is symmetric,...

Let A be an n × n real symmetric matrix with its row and column sums...

Q. Let A be a real n×n matrix. (a) Show that A =0 if AA^T =0....

Let A be a diagonalizable n × n matrix and let P be an invertible n...

Let A be a diagonalizable n × n matrix and let P be an invertible n...

Suppose C is a m × n matrix and A is a n × m matrix....

Let A be an m × n matrix and B be an m × p matrix....

let A be a real matrix of size MN.,assume that nullity (A^T A)=r. Find the range...

Let v1 be an eigenvector of an n×n matrix A corresponding to λ1, and let v2,...

Let A be an n × n matrix which is not 0 but A2 = 0....