Question

In: Advanced Math

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

Let A∈R^{n× n} be a non-symmetric matrix.

Prove that |λ₁| is real, provided that |λ₁|>|λ₂|≥|λ₃|≥...≥|λ_n| where λ_i , i= 1,...,n are the eigenvalues of A, while others can be real or not real.

Expert Solution

Consider the characteristic equation . The eigenvalues of the matrix are the roots of this characteristic equation.

Suppose that are the eigenvalues. Suppose that is not real; then for some reals with . Since is a polynomial equation with real coefficients, and since is a root of this polynomial, its conjugate is also a root of this polynomial. Thus, for some . Note that since ; however, .

Thus, whenever a complex eigenvalue occurs then its absolute value is attained by at least two distinct complex eigenvalues. In particular, if the eigenvalue is such that , then must be real.

To show that the other eigenvalues can be real or complex, we note that

and

are both non-symmetric; while the eigenvalues of are , those of are . Thus, the other eigenvalues can be real or complex.

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...

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

Let v1 be an eigenvector of an n×n matrix A corresponding to λ1, and let v2, v3 be two linearly independent eigenvectors of A corresponding to λ2, where λ1 is not equal to λ2. Show that v1, v2, v3 are linearly independent.

Let A ∈ Rn×n be symmetric and positive definite and let C ∈ Rn×m. Show:, rank(C^TAC)...

Let A ∈ Rn×n be symmetric and positive definite and let C ∈ Rn×m. Show:, rank(C^TAC) = rank(C),

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 W be a subspace of Rn. Prove that W⊥ is also a subspace of Rn.

Let A be an m x n matrix. Prove that Ax = b has at least...

Let A be an m x n matrix. Prove that Ax = b has at least one solution for any b if and only if A has linearly independent rows. Let V be a vector space with dimension 3, and let V = span(u, v, w). Prove that u, v, w are linearly independent (in other words, you are being asked to show that u, v, w form a basis for V)

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

For matrices A ∈ Rn×n and B ∈ Rn×p, prove each of the following statements: (a)...

For matrices A ∈ Rn×n and B ∈ Rn×p, prove each of the following statements: (a) rank(AB) = rank(A) and R(AB) = R(A) if rank(B) = n. (b) rank(AB) = rank(B) and N (AB) = N (B) if rank(A) = n.

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 =