IIT JAM & CSIR NET - Mathematical Sciences

Vector Spaces & Matrix Theory question bank (Linear Algebra & Real Analysis)

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Q1.
[Linear Algebra & Real Analysis Problem #1] Regarding eigenvalues of a real symmetric matrix A: Which of the following analytical formulations is correct?

(A)
The matrix cannot be diagonalized under any orthogonal transformation
(B)
All eigenvalues are pure imaginary
(C)
All eigenvalues are strictly real, and eigenvectors from distinct eigenvalues are orthogonal
(D)
Eigenvalues must always be positive integers
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Q2.
[Linear Algebra & Real Analysis Problem #2] Regarding eigenvalues of a real symmetric matrix A: Which of the following analytical formulations is correct?

(A)
All eigenvalues are strictly real, and eigenvectors from distinct eigenvalues are orthogonal
(B)
The matrix cannot be diagonalized under any orthogonal transformation
(C)
Eigenvalues must always be positive integers
(D)
All eigenvalues are pure imaginary
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