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Let us consider a square matrix A of order n with Eigen values of a, b, c then the Eigen values of the matrix AT could be.
Correct answer is 'a,b,c'. Can you explain this answer?
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Let us consider a square matrix A of order n with Eigen values of a, b...
According to the property of the Eigen values, any square matrix A and its transpose AT have the same Eigen values.
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Let us consider a square matrix A of order n with Eigen values of a, b...
Understanding Eigenvalues of a Matrix and its Transpose
When we have a square matrix A of order n, its eigenvalues represent the scalar values for which there exists a non-zero vector (the eigenvector) such that the equation A*v = λ*v holds, where λ is an eigenvalue and v is an eigenvector.
Eigenvalues and Transpose
- The transpose of a matrix, denoted as AT, is obtained by swapping its rows and columns.
- An essential property of matrices is that the eigenvalues of a matrix and its transpose are the same.
Proof Concept
- If λ is an eigenvalue of A, then there exists a non-zero vector v such that A*v = λ*v.
- Taking the transpose of this equation, we have (A*v)T = (λ*v)T.
- Using the property of transposes: (AB)T = BTAT, we can rewrite this as vT * AT = λ*vT.
- This shows that if v is an eigenvector of A corresponding to eigenvalue λ, then vT is an eigenvector of AT corresponding to the same eigenvalue λ.
Conclusion
- Therefore, for a square matrix A with eigenvalues a, b, and c, the eigenvalues of the transpose matrix AT are also a, b, and c.
- This result holds true for any square matrix, confirming the equality of eigenvalues for a matrix and its transpose.
In summary, the eigenvalues of the transpose matrix AT are the same as those of the original matrix A, which are a, b, and c.
When we have a square matrix A of order n, its eigenvalues represent the scalar values for which there exists a non-zero vector (the eigenvector) such that the equation A*v = λ*v holds, where λ is an eigenvalue and v is an eigenvector.
Eigenvalues and Transpose
- The transpose of a matrix, denoted as AT, is obtained by swapping its rows and columns.
- An essential property of matrices is that the eigenvalues of a matrix and its transpose are the same.
Proof Concept
- If λ is an eigenvalue of A, then there exists a non-zero vector v such that A*v = λ*v.
- Taking the transpose of this equation, we have (A*v)T = (λ*v)T.
- Using the property of transposes: (AB)T = BTAT, we can rewrite this as vT * AT = λ*vT.
- This shows that if v is an eigenvector of A corresponding to eigenvalue λ, then vT is an eigenvector of AT corresponding to the same eigenvalue λ.
Conclusion
- Therefore, for a square matrix A with eigenvalues a, b, and c, the eigenvalues of the transpose matrix AT are also a, b, and c.
- This result holds true for any square matrix, confirming the equality of eigenvalues for a matrix and its transpose.
In summary, the eigenvalues of the transpose matrix AT are the same as those of the original matrix A, which are a, b, and c.
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Let us consider a square matrix A of order n with Eigen values of a, b, c then the Eigen values of the matrix AT could be.Correct answer is 'a,b,c'. Can you explain this answer? for Mathematics 2025 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let us consider a square matrix A of order n with Eigen values of a, b, c then the Eigen values of the matrix AT could be.Correct answer is 'a,b,c'. Can you explain this answer? covers all topics & solutions for Mathematics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let us consider a square matrix A of order n with Eigen values of a, b, c then the Eigen values of the matrix AT could be.Correct answer is 'a,b,c'. Can you explain this answer?.
Let us consider a square matrix A of order n with Eigen values of a, b, c then the Eigen values of the matrix AT could be.Correct answer is 'a,b,c'. Can you explain this answer? for Mathematics 2025 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let us consider a square matrix A of order n with Eigen values of a, b, c then the Eigen values of the matrix AT could be.Correct answer is 'a,b,c'. Can you explain this answer? covers all topics & solutions for Mathematics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let us consider a square matrix A of order n with Eigen values of a, b, c then the Eigen values of the matrix AT could be.Correct answer is 'a,b,c'. Can you explain this answer?.
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