Mechanical Engineering Exam > Mechanical Engineering Questions > If a square matrix A is real and symmetric, t...
Start Learning for Free
If a square matrix A is real and symmetric, then the eigenvaluesn
- a)Are always real
- b)Are always real and positive
- c)Are always real and non-negative
- d)Occur in complex conjugate pairs
Correct answer is option 'A'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Verified Answer
If a square matrix A is real and symmetric, then the eigenvaluesna)Are...
Most Upvoted Answer
If a square matrix A is real and symmetric, then the eigenvaluesna)Are...
Real and Symmetric Square Matrix
A square matrix is a matrix with an equal number of rows and columns. In this case, we are considering a real and symmetric square matrix. A real matrix is a matrix that consists of real numbers as its elements, while a symmetric matrix is a matrix that is equal to its transpose.
Eigenvalues of a Matrix
Eigenvalues are a special set of scalars associated with a linear system of equations represented by a matrix. In other words, they are values that satisfy the equation Av = λv, where A is the matrix, λ is the eigenvalue, and v is the eigenvector.
Properties of Real and Symmetric Square Matrix
Real and symmetric square matrices have some specific properties that help determine the nature of their eigenvalues. These properties are:
1. Real Eigenvalues: Eigenvalues of a real matrix are always real numbers. This means that they do not have any imaginary components. Therefore, option 'a' is correct.
2. Orthogonal Eigenvectors: Real and symmetric matrices have orthogonal eigenvectors. Orthogonal vectors are perpendicular to each other, and their dot product is zero. This property helps in diagonalizing the matrix.
3. Diagonalization: Real and symmetric matrices can always be diagonalized. Diagonalization means transforming the matrix into a diagonal matrix by using the eigenvectors as transformation matrices. This process helps in simplifying calculations involving the matrix.
4. Positive Definiteness: Real and symmetric matrices can have positive eigenvalues, negative eigenvalues, or zero eigenvalues depending on their characteristics. However, it is not always true that the eigenvalues are positive. Therefore, option 'b' is incorrect.
5. Complex Conjugate Pairs: Complex conjugate pairs occur in matrices that are not real and symmetric. In the case of real and symmetric matrices, the eigenvalues are always real and do not occur in complex conjugate pairs. Therefore, option 'd' is incorrect.
Conclusion
In conclusion, if a square matrix is real and symmetric, its eigenvalues are always real. This is because real and symmetric matrices have specific properties that guarantee real eigenvalues. These properties include real eigenvalues, orthogonal eigenvectors, diagonalization, and the absence of complex conjugate pairs.
A square matrix is a matrix with an equal number of rows and columns. In this case, we are considering a real and symmetric square matrix. A real matrix is a matrix that consists of real numbers as its elements, while a symmetric matrix is a matrix that is equal to its transpose.
Eigenvalues of a Matrix
Eigenvalues are a special set of scalars associated with a linear system of equations represented by a matrix. In other words, they are values that satisfy the equation Av = λv, where A is the matrix, λ is the eigenvalue, and v is the eigenvector.
Properties of Real and Symmetric Square Matrix
Real and symmetric square matrices have some specific properties that help determine the nature of their eigenvalues. These properties are:
1. Real Eigenvalues: Eigenvalues of a real matrix are always real numbers. This means that they do not have any imaginary components. Therefore, option 'a' is correct.
2. Orthogonal Eigenvectors: Real and symmetric matrices have orthogonal eigenvectors. Orthogonal vectors are perpendicular to each other, and their dot product is zero. This property helps in diagonalizing the matrix.
3. Diagonalization: Real and symmetric matrices can always be diagonalized. Diagonalization means transforming the matrix into a diagonal matrix by using the eigenvectors as transformation matrices. This process helps in simplifying calculations involving the matrix.
4. Positive Definiteness: Real and symmetric matrices can have positive eigenvalues, negative eigenvalues, or zero eigenvalues depending on their characteristics. However, it is not always true that the eigenvalues are positive. Therefore, option 'b' is incorrect.
5. Complex Conjugate Pairs: Complex conjugate pairs occur in matrices that are not real and symmetric. In the case of real and symmetric matrices, the eigenvalues are always real and do not occur in complex conjugate pairs. Therefore, option 'd' is incorrect.
Conclusion
In conclusion, if a square matrix is real and symmetric, its eigenvalues are always real. This is because real and symmetric matrices have specific properties that guarantee real eigenvalues. These properties include real eigenvalues, orthogonal eigenvectors, diagonalization, and the absence of complex conjugate pairs.
Free Test
FREE
| Start Free Test |
Community Answer
If a square matrix A is real and symmetric, then the eigenvaluesna)Are...
A
Attention Mechanical Engineering Students!
To make sure you are not studying endlessly, EduRev has designed Mechanical Engineering study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Mechanical Engineering.
|
Explore Courses for Mechanical Engineering exam
|
|
Similar Mechanical Engineering Doubts
Top Courses for Mechanical EngineeringView all
If a square matrix A is real and symmetric, then the eigenvaluesna)Are always real b)Are always real and positivec)Are always real and non-negatived)Occur in complex conjugate pairsCorrect answer is option 'A'. Can you explain this answer?
Question Description
If a square matrix A is real and symmetric, then the eigenvaluesna)Are always real b)Are always real and positivec)Are always real and non-negatived)Occur in complex conjugate pairsCorrect answer is option 'A'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about If a square matrix A is real and symmetric, then the eigenvaluesna)Are always real b)Are always real and positivec)Are always real and non-negatived)Occur in complex conjugate pairsCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If a square matrix A is real and symmetric, then the eigenvaluesna)Are always real b)Are always real and positivec)Are always real and non-negatived)Occur in complex conjugate pairsCorrect answer is option 'A'. Can you explain this answer?.
If a square matrix A is real and symmetric, then the eigenvaluesna)Are always real b)Are always real and positivec)Are always real and non-negatived)Occur in complex conjugate pairsCorrect answer is option 'A'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about If a square matrix A is real and symmetric, then the eigenvaluesna)Are always real b)Are always real and positivec)Are always real and non-negatived)Occur in complex conjugate pairsCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If a square matrix A is real and symmetric, then the eigenvaluesna)Are always real b)Are always real and positivec)Are always real and non-negatived)Occur in complex conjugate pairsCorrect answer is option 'A'. Can you explain this answer?.
Solutions for If a square matrix A is real and symmetric, then the eigenvaluesna)Are always real b)Are always real and positivec)Are always real and non-negatived)Occur in complex conjugate pairsCorrect answer is option 'A'. Can you explain this answer? in English & in Hindi are available as part of our courses for Mechanical Engineering.
Download more important topics, notes, lectures and mock test series for Mechanical Engineering Exam by signing up for free.
Here you can find the meaning of If a square matrix A is real and symmetric, then the eigenvaluesna)Are always real b)Are always real and positivec)Are always real and non-negatived)Occur in complex conjugate pairsCorrect answer is option 'A'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
If a square matrix A is real and symmetric, then the eigenvaluesna)Are always real b)Are always real and positivec)Are always real and non-negatived)Occur in complex conjugate pairsCorrect answer is option 'A'. Can you explain this answer?, a detailed solution for If a square matrix A is real and symmetric, then the eigenvaluesna)Are always real b)Are always real and positivec)Are always real and non-negatived)Occur in complex conjugate pairsCorrect answer is option 'A'. Can you explain this answer? has been provided alongside types of If a square matrix A is real and symmetric, then the eigenvaluesna)Are always real b)Are always real and positivec)Are always real and non-negatived)Occur in complex conjugate pairsCorrect answer is option 'A'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice If a square matrix A is real and symmetric, then the eigenvaluesna)Are always real b)Are always real and positivec)Are always real and non-negatived)Occur in complex conjugate pairsCorrect answer is option 'A'. Can you explain this answer? tests, examples and also practice Mechanical Engineering tests.
|
|
Explore Courses for Mechanical Engineering exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup