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The modulus of eigenvalues of an orthogonal matrix is
- a)0
- b)-1
- c)1
- d)2
Correct answer is option 'C'. Can you explain this answer?
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The modulus of eigenvalues of an orthogonal matrix isa)0b)-1c)1d)2Corr...
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Modulus of Eigenvalues of an Orthogonal Matrix
An orthogonal matrix is a square matrix where the columns and rows are orthonormal vectors, meaning they are unit vectors that are orthogonal to each other. The modulus of eigenvalues of an orthogonal matrix is always 1. Let's break this down further:
Orthogonal Matrix Properties
- Orthogonal matrices have the property \(Q^TQ = I\), where \(Q^T\) is the transpose of the matrix Q and I is the identity matrix.
- The eigenvalues of an orthogonal matrix are always complex numbers with unit modulus, meaning their absolute value is always 1.
Understanding Eigenvalues of Orthogonal Matrix
- Let λ be an eigenvalue of an orthogonal matrix Q with corresponding eigenvector x.
- By the definition of eigenvalues, we have \(Qx = \lambda x\).
- Taking the modulus of both sides, we get \(|Qx| = |\lambda x|\).
- Since Q is an orthogonal matrix, the modulus of x remains the same after multiplication by Q. Thus, \(|x| = |\lambda x|\).
- This implies that \(|\lambda| = 1\), as the modulus of x is always 1 for unit vectors.
Therefore, the modulus of eigenvalues of an orthogonal matrix is always 1.
An orthogonal matrix is a square matrix where the columns and rows are orthonormal vectors, meaning they are unit vectors that are orthogonal to each other. The modulus of eigenvalues of an orthogonal matrix is always 1. Let's break this down further:
Orthogonal Matrix Properties
- Orthogonal matrices have the property \(Q^TQ = I\), where \(Q^T\) is the transpose of the matrix Q and I is the identity matrix.
- The eigenvalues of an orthogonal matrix are always complex numbers with unit modulus, meaning their absolute value is always 1.
Understanding Eigenvalues of Orthogonal Matrix
- Let λ be an eigenvalue of an orthogonal matrix Q with corresponding eigenvector x.
- By the definition of eigenvalues, we have \(Qx = \lambda x\).
- Taking the modulus of both sides, we get \(|Qx| = |\lambda x|\).
- Since Q is an orthogonal matrix, the modulus of x remains the same after multiplication by Q. Thus, \(|x| = |\lambda x|\).
- This implies that \(|\lambda| = 1\), as the modulus of x is always 1 for unit vectors.
Therefore, the modulus of eigenvalues of an orthogonal matrix is always 1.
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The modulus of eigenvalues of an orthogonal matrix isa)0b)-1c)1d)2Correct answer is option 'C'. Can you explain this answer?
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The modulus of eigenvalues of an orthogonal matrix isa)0b)-1c)1d)2Correct answer is option 'C'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about The modulus of eigenvalues of an orthogonal matrix isa)0b)-1c)1d)2Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The modulus of eigenvalues of an orthogonal matrix isa)0b)-1c)1d)2Correct answer is option 'C'. Can you explain this answer?.
The modulus of eigenvalues of an orthogonal matrix isa)0b)-1c)1d)2Correct answer is option 'C'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about The modulus of eigenvalues of an orthogonal matrix isa)0b)-1c)1d)2Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The modulus of eigenvalues of an orthogonal matrix isa)0b)-1c)1d)2Correct answer is option 'C'. Can you explain this answer?.
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