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 or 1.

(d) Set V1 = {v ∈ C n | Av = v} and V0 = {v ∈ C n | Av = 0}. Show that im A = V1 and ker A = V0.

(e) Prove that A is diagonalizable.

Expert Solution

Colby Messinger answered 2 years ago

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...

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.

Let A∈Rn× n be a non-symmetric matrix. Prove that |λ1| is real, provided that |λ1|>|λ2|≥|λ3|≥...≥|λn| where...

Let A∈Rn× n be a non-symmetric matrix. Prove that |λ1| is real, provided that |λ1|>|λ2|≥|λ3|≥...≥|λn| where λi , i= 1,...,n are the eigenvalues of A, while others can be real or not real.

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

let A be a real matrix of size M*N.,assume that nullity (A^T* A)=r. Find the range of values that 'r' can take using values of 'M' and 'N'. also find the nullity (A^T)

A square matrix A is said to be symmetric if its transpose AT satisfies AT= A,...

A square matrix A is said to be symmetric if its transpose AT satisfies AT= A, and a complex-valued square matrix A is said to be Hermitian if its conjugate transpose AH = (A)T = AT satisfies AH = A. Thus, a real-valued square matrix A is symmetric if and only if it is Hermitian. Which of the following is a vector space? (a) The set of all n xn real-valued symmetric matrices over R. (b) The set of all...

Let A be a m × n matrix with entries in R. Recall that the row...

Let A be a m × n matrix with entries in R. Recall that the row rank of A means the dimension of the subspace in RN spanned by the rows of A (viewed as vectors in Rn), and the column rank means that of the subspace in Rm spanned by the columns of A (viewed as vectors in Rm). (a) Prove that n = (column rank of A) + dim S, where the set S is the solution space...

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

A JAVA program that will read a boolean matrix corresponding to a relation R and output whether R is Reflexive, Symmetric, Anti-Symmetric and/or Transitive.

A JAVA program that will read a boolean matrix corresponding to a relation R and output whether R is Reflexive, Symmetric, Anti-Symmetric and/or Transitive. Input to the program will be the size n of an n x n boolean matrix followed by the matrix elements. Document your program nicely.NOTE: The program must output a reason in the case that an input relation fails to have a certain property.

Question

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

Solutions

Expert Solution

Related Solutions

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

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

Let A∈Rn× n be a non-symmetric matrix. Prove that |λ1| is real, provided that |λ1|>|λ2|≥|λ3|≥...≥|λn| where...

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

A square matrix A is said to be symmetric if its transpose AT satisfies AT= A,...

Let A be a m × n matrix with entries in R. Recall that the row...

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...

A JAVA program that will read a boolean matrix corresponding to a relation R and output whether R is Reflexive, Symmetric, Anti-Symmetric and/or Transitive.