I understand how certain matrices can be orthonormal and the conditions necessary for that. I don't understand its geometric relevance. For instance in a vector space the axis would be orthonormal and perhaps another set of matrices, but so what ?
2026-03-26 01:28:27.1774488507
Do orthonormal matrices have a geometric interpretation or contextual importance?
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When $M$ is orthonormal and $u,v$ are vectors,the distance from $u$ to $v$ equals the distance from $Mu$ to $Mv$. So the function that maps each $u$ to $Mu$ does not change angles or shapes. Such a function can be interpreted geometrically as a rotation, keeping the origin fixed, or a rotation followed by a reflection in a mirror (still keeping the origin fixed).