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Q.No. 11 – 35 Carry One Mark Each
Which one of the following statements is true for all real symmetric matrices?
- a)All the eigenvalues are real.
- b)All the eigenvalues are positive
- c)All the eigenvalues are distinct
- d)Sum of all the eigenvalues is zero.
Correct answer is option 'A'. Can you explain this answer?
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Q.No. 11 – 35 Carry One Mark EachWhich one of the following stat...
Eigen values of a real symmetric matrix are all real
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Q.No. 11 – 35 Carry One Mark EachWhich one of the following stat...
Eigenvalues of Real Symmetric Matrices
Real symmetric matrices have unique properties that make them an important topic in linear algebra. Among the options provided, the statement that all eigenvalues are real holds true for all real symmetric matrices.
Key Points about Real Symmetric Matrices:
- Real Eigenvalues:
- A fundamental theorem states that if a matrix is symmetric (i.e., it equals its transpose), then all its eigenvalues are guaranteed to be real numbers. This is a consequence of the Spectral Theorem.
- Positive Eigenvalues:
- While it's true that symmetric matrices can have positive eigenvalues, they can also have negative or zero eigenvalues. Therefore, this statement is not universally true.
- Distinct Eigenvalues:
- Symmetric matrices can have repeated eigenvalues. Thus, this statement does not hold for all real symmetric matrices.
- Sum of Eigenvalues:
- The sum of the eigenvalues equals the trace of the matrix, which is not necessarily zero for all matrices.
Conclusion
Thus, the only universally applicable statement for all real symmetric matrices is that all eigenvalues are real. This property is crucial in various applications, including stability analysis and quadratic forms, making it a foundational aspect of matrix theory.
Real symmetric matrices have unique properties that make them an important topic in linear algebra. Among the options provided, the statement that all eigenvalues are real holds true for all real symmetric matrices.
Key Points about Real Symmetric Matrices:
- Real Eigenvalues:
- A fundamental theorem states that if a matrix is symmetric (i.e., it equals its transpose), then all its eigenvalues are guaranteed to be real numbers. This is a consequence of the Spectral Theorem.
- Positive Eigenvalues:
- While it's true that symmetric matrices can have positive eigenvalues, they can also have negative or zero eigenvalues. Therefore, this statement is not universally true.
- Distinct Eigenvalues:
- Symmetric matrices can have repeated eigenvalues. Thus, this statement does not hold for all real symmetric matrices.
- Sum of Eigenvalues:
- The sum of the eigenvalues equals the trace of the matrix, which is not necessarily zero for all matrices.
Conclusion
Thus, the only universally applicable statement for all real symmetric matrices is that all eigenvalues are real. This property is crucial in various applications, including stability analysis and quadratic forms, making it a foundational aspect of matrix theory.
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Q.No. 11 – 35 Carry One Mark EachWhich one of the following statements is true for all real symmetric matrices?a)All the eigenvalues are real.b)All the eigenvalues are positivec)All the eigenvalues are distinctd)Sum of all the eigenvalues is zero.Correct answer is option 'A'. Can you explain this answer?
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Q.No. 11 – 35 Carry One Mark EachWhich one of the following statements is true for all real symmetric matrices?a)All the eigenvalues are real.b)All the eigenvalues are positivec)All the eigenvalues are distinctd)Sum of all the eigenvalues is zero.Correct answer is option 'A'. 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 Q.No. 11 – 35 Carry One Mark EachWhich one of the following statements is true for all real symmetric matrices?a)All the eigenvalues are real.b)All the eigenvalues are positivec)All the eigenvalues are distinctd)Sum of all the eigenvalues is zero.Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Q.No. 11 – 35 Carry One Mark EachWhich one of the following statements is true for all real symmetric matrices?a)All the eigenvalues are real.b)All the eigenvalues are positivec)All the eigenvalues are distinctd)Sum of all the eigenvalues is zero.Correct answer is option 'A'. Can you explain this answer?.
Q.No. 11 – 35 Carry One Mark EachWhich one of the following statements is true for all real symmetric matrices?a)All the eigenvalues are real.b)All the eigenvalues are positivec)All the eigenvalues are distinctd)Sum of all the eigenvalues is zero.Correct answer is option 'A'. 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 Q.No. 11 – 35 Carry One Mark EachWhich one of the following statements is true for all real symmetric matrices?a)All the eigenvalues are real.b)All the eigenvalues are positivec)All the eigenvalues are distinctd)Sum of all the eigenvalues is zero.Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Q.No. 11 – 35 Carry One Mark EachWhich one of the following statements is true for all real symmetric matrices?a)All the eigenvalues are real.b)All the eigenvalues are positivec)All the eigenvalues are distinctd)Sum of all the eigenvalues is zero.Correct answer is option 'A'. Can you explain this answer?.
Solutions for Q.No. 11 – 35 Carry One Mark EachWhich one of the following statements is true for all real symmetric matrices?a)All the eigenvalues are real.b)All the eigenvalues are positivec)All the eigenvalues are distinctd)Sum of all the eigenvalues is zero.Correct answer is option 'A'. Can you explain this answer? in English & in Hindi are available as part of our courses for GATE.
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Q.No. 11 – 35 Carry One Mark EachWhich one of the following statements is true for all real symmetric matrices?a)All the eigenvalues are real.b)All the eigenvalues are positivec)All the eigenvalues are distinctd)Sum of all the eigenvalues is zero.Correct answer is option 'A'. Can you explain this answer?, a detailed solution for Q.No. 11 – 35 Carry One Mark EachWhich one of the following statements is true for all real symmetric matrices?a)All the eigenvalues are real.b)All the eigenvalues are positivec)All the eigenvalues are distinctd)Sum of all the eigenvalues is zero.Correct answer is option 'A'. Can you explain this answer? has been provided alongside types of Q.No. 11 – 35 Carry One Mark EachWhich one of the following statements is true for all real symmetric matrices?a)All the eigenvalues are real.b)All the eigenvalues are positivec)All the eigenvalues are distinctd)Sum of all the eigenvalues is zero.Correct answer is option 'A'. Can you explain this answer? theory, EduRev gives you an
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