VI. TRUE or FALSE. Justify your answers concisely.1. If A € M₂ (C) then A* - A is Hermitian.2. Every eigenvector v of A € M₂ (R) is also an eigenvector of A².3. If A € M₂ (C) is skew-Hermitian then kA is also skew-Hermitian for k € C.

As per the question we are given the following statements and we have to justify whether they are…

Answered: TRUE or FALSE. Justify your answe swers…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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please send handwritten solution for Q1
Just answer it directly without explaing. Thanks.
Please show all steps clearly, thank you.
is an eigen-ket of the Hamiltonian and E is its eigen-value, the bra-vector of the 5. If quantum object can be written as E v) =exp-i- A. E v) =cxp| В. 2E v) =exp| C. exp D. E. None of the above Note: r is the space coordinate and t is the time.
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b. Which of the following statements are true? Select all that apply. A. The matrices A and AT have the same eigenvalues, counting multiplicities. B. The sum of two eigenvectors of a matrix A is also an eigenvector of A. | C. Each eigenvalue of A is also an eigenvalue of A?. D. Each eigenvector of A is also an eigenvector of A?. E. Each eigenvector of an invertible matrix A is also an eigenvector of A1.
3. For nx n matrix A. define R₁ = one of the Gerschgorin's disks {z: True False "=1,i+j lajj 1, then each eigenvalue of A is either in aij | ≤ Ri}. z-
Write a formal proof for the given statements below. (b) Let A be an n × n matrix. Suppose that vector v ∈ Rn is an eigenvector of A with the corresponding eigenvalue λ ∈ R. Prove that vector v is an eigenvector of A3 with the corresponding eigenvalue λ3 .
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22. Let P be an idempotent matrix. Show that the only eigenvalues of P are 1 Suppose that Px = Ax, x + 0.] 0 and 1 = 1. [Hint:

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TRUE or FALSE. Justify your answe swers concisely. 1. If A € M₂, (C) then A* - A is Hermitian. 2. Every eigenvector v of A € M₁ (R) is also an eigenvector of A². 3. If A € M₂ (C) is skew-Hermitian then kA is also skew-Hermitian for k € C.