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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?
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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.
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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?
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