GATE DA 2026 Set 1 — Question 65
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Linear Algebra → Eigenvalues & Eigenvectors → Eigenvalue Computation
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Question
Let be a matrix, where and is the identity matrix of order .The value of , where , is ________ .
(Answer in integer)
(Answer in integer)
Correct answer
1 to 1
Solution
The matrix is the centering matrix (often denoted or ). It is a projection matrix that projects vectors onto the subspace orthogonal to (the subspace of vectors with zero mean).The eigenvalues of the matrix are (with eigenvector ) and (with multiplicity ).
The eigenvalues of are and (multiplicity ).
The eigenvalues of are and (multiplicity ).The quadratic form subject to is maximized by the largest eigenvalue of (Rayleigh quotient).
.Thus, the maximum value is 1.
The eigenvalues of are and (multiplicity ).
The eigenvalues of are and (multiplicity ).The quadratic form subject to is maximized by the largest eigenvalue of (Rayleigh quotient).
.Thus, the maximum value is 1.
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