GATE DA 2024 Set 1 — Question 61
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Linear Algebra → Singular Value Decomposition → Singular Values
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Question
Let , and let be the singular values of the matrix (where is the transpose of ). The value of is ______.
Correct answer
55 to 55
Solution
The matrix is a rank-1 matrix formed by the outer product of the vector .
The non-zero eigenvalue of a rank-1 matrix is given by the inner product .Calculating the inner product:
The eigenvalues of are .
Since is a symmetric positive semi-definite matrix (as ), its singular values are equal to its eigenvalues.Thus, the singular values are .
The sum of the singular values is .
The non-zero eigenvalue of a rank-1 matrix is given by the inner product .Calculating the inner product:
The eigenvalues of are .
Since is a symmetric positive semi-definite matrix (as ), its singular values are equal to its eigenvalues.Thus, the singular values are .
The sum of the singular values is .
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