GATE 2022 PH – Question 13

Mathematical Physics · matrices: similarity transformations, diagonalization, eigenvalues and eigen vectors · 1 mark · Multiple choice

What is the maximum number of free independent real parameters specifying an n-dimensional orthogonal matrix?

$n(n-2)$
$(n-1)^2$
$n(n-1)/2$
$n(n+1)/2$

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Correct answer: (C) $n(n-1)/2$

Explanation

An $n\times n$ real matrix has $n^2$ independent entries.

**Orthogonality condition** $O^TO=I$ is a symmetric matrix equation. A symmetric $n\times n$ matrix has $\dfrac{n(n+1)}2$ independent entries, so the condition gives $\dfrac{n(n+1)}2$ independent constraints (the $n$ normalisation conditions and the $\dfrac{n(n-1)}2$ orthogonality conditions between different columns).

**Free parameters:**
$$n^2-\frac{n(n+1)}{2}=\frac{2n^2-n^2-n}{2}=\frac{n(n-1)}{2}\quad(\text{option C}).$$

(For $n=2$ this is 1 parameter, the rotation angle; for $n=3$ it is 3, the Euler angles.)