GATE 2026 DA – Question 65
Let $A = \left(I_n - \frac{1}{n}\mathbf{1}\mathbf{1}^T\right)$ be a matrix, where $\mathbf{1} = (1, 1, 1, \ldots, 1)^T \in \mathbb{R}^n$ and $I_n$ is the identity matrix of order $n$.
The value of $\max_{x \in S} x^TAx$, where $S = \{x \in \mathbb{R}^n \mid x^Tx = 1\}$, is __________ . (*Answer in integer*)
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: 1
Explanation
$A$ is symmetric, so the maximum of $x^TAx$ over unit vectors is its largest eigenvalue. $A$ is a projection matrix, so its eigenvalues are only 0 and 1: 0 for the direction $\mathbf{1}$ and 1 for every direction perpendicular to it. So the maximum is 1.