Discussion
Suppose that the columns of an \(n \times n\)-matrix M over \(\mathbf{R}\) are orthonormal. Then which of the following is not true ?
Answer: D
The statement that is not true is M is symmetric.
Orthonormal columns mean \(M^{T}M = I\), so M is orthogonal. Then \(\langle Mx, My\rangle = x^{T}M^{T}My = x^{T}y\), which gives option B, and with \(x = y\) it gives \(\|Mx\| = \|x\|\) (option A). For a square matrix \(M^{T}M = I\) forces \(MM^{T} = I\), so the rows are orthonormal too (option C).
An orthogonal matrix need not be symmetric. The rotation by 90°, with rows (0, -1) and (1, 0), has orthonormal columns, but its transpose is the rotation the other way, not itself.
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