Linear Algebra and Its Applications · Problem 3.4.6

Orthogonality — Gilbert Strang

Mathematics · orthogonal-matrix-third-column

Problem

Find a third column so that the matrix Q=[1/31/141/32/141/3−3/14]Q = \begin{bmatrix} 1/\sqrt{3} & 1/\sqrt{14} \\ 1/\sqrt{3} & 2/\sqrt{14} \\ 1/\sqrt{3} & -3/\sqrt{14} \end{bmatrix} is orthogonal. It must be a unit vector that is orthogonal to the other columns; how much freedom does this leave? Verify that the rows automatically become orthonormal at the same time.

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