GATE DA 2025 Set 1 — Question 50
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Linear Algebra → Eigenvalues & Eigenvectors → Determinant Properties
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Question
Let be a system of orthonormal vectors in . Consider the matrix . Which of the following statements is/are correct?
Correct answer
(A) Singular values of A are also its eigenvalues; (B) Singular values of A are either 0 or 1
Solution
The matrix is the orthogonal projection matrix onto the subspace spanned by the orthonormal vectors .
Since is a projection matrix, it is symmetric () and idempotent ().
The eigenvalues of a projection matrix are either 0 or 1. Since the rank of is 5 (sum of 5 rank-1 independent components), it has five eigenvalues equal to 1 and five equal to 0.
Since is symmetric and positive semi-definite (eigenvalues are non-negative), its singular values are equal to its eigenvalues.
Thus, singular values are 0 and 1. Option (A) and (B) are correct.
The determinant is the product of eigenvalues, which is 0. Thus, (C) is incorrect.
Since the determinant is 0, is not invertible. Thus, (D) is incorrect.
Since is a projection matrix, it is symmetric () and idempotent ().
The eigenvalues of a projection matrix are either 0 or 1. Since the rank of is 5 (sum of 5 rank-1 independent components), it has five eigenvalues equal to 1 and five equal to 0.
Since is symmetric and positive semi-definite (eigenvalues are non-negative), its singular values are equal to its eigenvalues.
Thus, singular values are 0 and 1. Option (A) and (B) are correct.
The determinant is the product of eigenvalues, which is 0. Thus, (C) is incorrect.
Since the determinant is 0, is not invertible. Thus, (D) is incorrect.
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