For Orthogonal Unit Vectors, Prove that Let $A$ be an $n \times n$ matrix. Prove that for $n\geq 2$ there exist $n$ orthogonal unit vectors $u_1,\ldots,u_n$ in $\mathbb{R^n}$ such that $Au_1,\ldots,Au_n$ are also orthogonal.
As I recall this is a theorem but couldnt find its proof. Can someone help me please.
This seems just a direct consequence of the Singular Value Decomposition for any matrix. In fact, $A=UDV$, where $D$ is diagonal and $U,V$ are orthogonal. As a consequence, $AV^T = UD$, but the columns of both $V^T$ and $UD$ are orthogonal.
This seems just a direct consequence of the Singular Value Decomposition for any matrix. In fact, $A=UDV$, where $D$ is diagonal and $U,V$ are orthogonal. As a consequence, $AV^T = UD$, but the columns of both $V^T$ and $UD$ are orthogonal.