Show that the set of all n × n real symmetric matrices with zero diagonal entries...

Show that the set of all n × n real symmetric matrices with zero diagonal entries is a subspace
of R^n×n

Expert Solution

Colby Messinger answered 2 years ago

P.0.2 Show that (a) the diagonal entries of a Hermitian matrix are real; (b) the diagonal...

P.0.2 Show that (a) the diagonal entries of a Hermitian matrix are real; (b) the diagonal entries of a skew-Hermitian matrix purely imaginary; c) the diagonal entries of a skew-symmetric matrix are zero. P.0.5 Let A ∈ Mn be invertible. Use mathematical induction to prove that (A-1)k = (Ak)-1 for all integers k. P.0.25 Let A ∈ Mn be idempotent. Show that A is invertible if and only if A = I P.0.26 Let A,B ∈ Mn be idempotent. Show...

(a) Show that the diagonal entries of a positive definite matrix are positive numbers. (b) Show...

(a) Show that the diagonal entries of a positive definite matrix are positive numbers. (b) Show that if B is a nonsingular square matrix, then BTB is an SPD matrix.(Hint. you simply need to show the positive definiteness, which does requires the nonsingularity of B.)

Determine whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric,...

Determine whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where (x,y) ∈ R if and only if: a) x = 1 OR y = 1 b) x = 1 I was curious about how those two compare. I have the solutions for part a) already.

Give an example of a square matrix A such that all the diagonal entries are positive...

Give an example of a square matrix A such that all the diagonal entries are positive but A is not positive definite

Let H be the subset of all skew-symmetric matrices in M3x3 a.) prove that H is...

Let H be the subset of all skew-symmetric matrices in M3x3 a.) prove that H is a subspace of M3x3 by checking all three conditions in the definition of subspace. b.) Find a basis for H. Prove that your basis is actually a basis for H by showing it is both linearly independent and spans H. c.) what is the dim(H)

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...

Show that two m×n matrices are equivalent if and only if they have the same invariant...

Show that two m×n matrices are equivalent if and only if they have the same invariant factors, i.e. (by Problem 4), if and only if they have the same Smith normal form.

If R is the 2×2 matrices over the real, show that R has nontrivial left and...

If R is the 2×2 matrices over the real, show that R has nontrivial left and right ideals.? hello could you please solve this problem with the clear hands writing to read it please? Also the good explanation to understand the solution is by step by step please thank the subject is Modern Algebra

Let R be the ring of all 2 x 2 real matrices. A. Assume that A...

Let R be the ring of all 2 x 2 real matrices. A. Assume that A is an element of R such that AB=BA for all B elements of R. Prove that A is a scalar multiple of the identity matrix. B. Prove that {0} and R are the only two ideals. Hint: Use the Matrices E11, E12, E21, E22.

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