= Orthogonal change of coordinates
{title2=$x=Qy,\quad Q^TQ=I$}
An orthogonal change of coordinates relates two <orthonormal bases> of a real <inner product space>. With the new basis vectors as columns of an <orthogonal matrix> $Q$, old and new coordinate columns obey $x=Qy$. Then $x^Tx=y^Ty$, so <Euclidean norm>, <Euclidean distance> and <angles> are preserved. A <quadratic form> $x^TAx$ becomes $y^T(Q^TAQ)y$. For a <symmetric matrix>, <orthogonal diagonalization> chooses $Q$ so that the transformed quadratic form is a sum of <eigenvalues> times squared coordinates. This explains the principal-axis descriptions of a <quadric surface>.
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