The GATE Grind

GATE 2026 MA – Question 32

Functional Analysis · Bounded Linear Operators and Duality · 1 mark · Numerical answer

On ℝ³ with ℓ¹ norm, T has matrix $\begin{pmatrix}1&1&3\\2&2&-2\\1&3&-3\end{pmatrix}$. Find its operator norm (integer).

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Correct answer: 8 to 8

Explanation

The induced ℓ¹ norm is maximum absolute column sum. These sums are 1+2+1 = 4, 1+2+3 = 6 and 3+2+3 = 8. The bound is attained at the third coordinate unit vector. **‖T‖ = 8**.

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