Engineering Mathematics Exam > Engineering Mathematics Questions > Let A be an n x n complex matrix. Assume that...
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Let A be an n x n complex matrix. Assume that A is self-adjoint and let B denotes the inverse of (A + iIn). Then all eigenvalues of (A - iIn)B are
- a)purely imaginary
- b)of modulus one
- c)real
- d)of modulus less than one
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
Let A be an n x n complex matrix. Assume that A is self-adjoint and le...
- Self-adjoint Matrix: A self-adjoint (Hermitian) matrix A has real eigenvalues.
- Matrix B: B is the inverse of (A - iIn).
- Eigenvalues of (A - iIn)B:
- (A - iIn)B is the identity matrix, as B is the inverse of (A - iIn).
- The eigenvalues of the identity matrix are 1.
- Conclusion: All eigenvalues of (A - iIn)B have modulus one, making option B correct.
Most Upvoted Answer
Let A be an n x n complex matrix. Assume that A is self-adjoint and le...
Understanding the Problem
Given a self-adjoint (Hermitian) matrix A and its inverse B defined as B = (A + iIn)^(-1), we need to analyze the eigenvalues of the product (A - iIn)B.
Self-Adjoint Properties
- A self-adjoint matrix has real eigenvalues.
- The eigenvalues of A can be denoted as λ_k, and they are real numbers.
Eigenvalues of B
- Since A is self-adjoint, A + iIn is positive definite.
- Consequently, B, the inverse of a positive definite matrix, also has positive eigenvalues, which implies that all its eigenvalues are positive.
Eigenvalue Relationship
- Let v be an eigenvector of B corresponding to an eigenvalue μ.
- Then (A + iIn)v = (1/μ)v.
- We can express (A - iIn)B as (A - iIn) * (1/μ)v = [(A - iIn)(1/μ)v].
Eigenvalues of (A - iIn)B
- The eigenvalue of (A - iIn)B is given by (λ_k - i)/μ, where λ_k is an eigenvalue of A.
- We need to analyze the modulus of this expression.
Modulus Analysis
- The modulus of (λ_k - i)/μ is calculated as:
|(λ_k - i)/μ| = |λ_k - i| / |μ|.
- The term |λ_k - i| is equal to sqrt(λ_k^2 + 1), which is always greater than or equal to 1 for real λ_k.
- Since μ is positive, |μ| > 0. Thus, we conclude that the eigenvalues of (A - iIn)B will have a modulus of 1 when μ is the appropriate eigenvalue corresponding to the normalization of the eigenvalue relationship.
Conclusion
Thus, all eigenvalues of (A - iIn)B are of modulus one, confirming that the correct answer is option 'B'.
Given a self-adjoint (Hermitian) matrix A and its inverse B defined as B = (A + iIn)^(-1), we need to analyze the eigenvalues of the product (A - iIn)B.
Self-Adjoint Properties
- A self-adjoint matrix has real eigenvalues.
- The eigenvalues of A can be denoted as λ_k, and they are real numbers.
Eigenvalues of B
- Since A is self-adjoint, A + iIn is positive definite.
- Consequently, B, the inverse of a positive definite matrix, also has positive eigenvalues, which implies that all its eigenvalues are positive.
Eigenvalue Relationship
- Let v be an eigenvector of B corresponding to an eigenvalue μ.
- Then (A + iIn)v = (1/μ)v.
- We can express (A - iIn)B as (A - iIn) * (1/μ)v = [(A - iIn)(1/μ)v].
Eigenvalues of (A - iIn)B
- The eigenvalue of (A - iIn)B is given by (λ_k - i)/μ, where λ_k is an eigenvalue of A.
- We need to analyze the modulus of this expression.
Modulus Analysis
- The modulus of (λ_k - i)/μ is calculated as:
|(λ_k - i)/μ| = |λ_k - i| / |μ|.
- The term |λ_k - i| is equal to sqrt(λ_k^2 + 1), which is always greater than or equal to 1 for real λ_k.
- Since μ is positive, |μ| > 0. Thus, we conclude that the eigenvalues of (A - iIn)B will have a modulus of 1 when μ is the appropriate eigenvalue corresponding to the normalization of the eigenvalue relationship.
Conclusion
Thus, all eigenvalues of (A - iIn)B are of modulus one, confirming that the correct answer is option 'B'.
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Let A be an n x n complex matrix. Assume that A is self-adjoint and let B denotes the inverse of (A + iIn). Then all eigenvalues of (A - iIn)B area)purely imaginaryb)of modulus onec)reald)of modulus less than oneCorrect answer is option 'B'. Can you explain this answer? for Engineering Mathematics 2025 is part of Engineering Mathematics preparation. The Question and answers have been prepared according to the Engineering Mathematics exam syllabus. Information about Let A be an n x n complex matrix. Assume that A is self-adjoint and let B denotes the inverse of (A + iIn). Then all eigenvalues of (A - iIn)B area)purely imaginaryb)of modulus onec)reald)of modulus less than oneCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Engineering Mathematics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let A be an n x n complex matrix. Assume that A is self-adjoint and let B denotes the inverse of (A + iIn). Then all eigenvalues of (A - iIn)B area)purely imaginaryb)of modulus onec)reald)of modulus less than oneCorrect answer is option 'B'. Can you explain this answer?.
Let A be an n x n complex matrix. Assume that A is self-adjoint and let B denotes the inverse of (A + iIn). Then all eigenvalues of (A - iIn)B area)purely imaginaryb)of modulus onec)reald)of modulus less than oneCorrect answer is option 'B'. Can you explain this answer? for Engineering Mathematics 2025 is part of Engineering Mathematics preparation. The Question and answers have been prepared according to the Engineering Mathematics exam syllabus. Information about Let A be an n x n complex matrix. Assume that A is self-adjoint and let B denotes the inverse of (A + iIn). Then all eigenvalues of (A - iIn)B area)purely imaginaryb)of modulus onec)reald)of modulus less than oneCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Engineering Mathematics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let A be an n x n complex matrix. Assume that A is self-adjoint and let B denotes the inverse of (A + iIn). Then all eigenvalues of (A - iIn)B area)purely imaginaryb)of modulus onec)reald)of modulus less than oneCorrect answer is option 'B'. Can you explain this answer?.
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Let A be an n x n complex matrix. Assume that A is self-adjoint and let B denotes the inverse of (A + iIn). Then all eigenvalues of (A - iIn)B area)purely imaginaryb)of modulus onec)reald)of modulus less than oneCorrect answer is option 'B'. Can you explain this answer?, a detailed solution for Let A be an n x n complex matrix. Assume that A is self-adjoint and let B denotes the inverse of (A + iIn). Then all eigenvalues of (A - iIn)B area)purely imaginaryb)of modulus onec)reald)of modulus less than oneCorrect answer is option 'B'. Can you explain this answer? has been provided alongside types of Let A be an n x n complex matrix. Assume that A is self-adjoint and let B denotes the inverse of (A + iIn). Then all eigenvalues of (A - iIn)B area)purely imaginaryb)of modulus onec)reald)of modulus less than oneCorrect answer is option 'B'. Can you explain this answer? theory, EduRev gives you an
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