The Victory of Orthogonality

The equation $A x=0$ tells us that x is perpendicular to every row of $A-$ and therefore to the whole row space of A. This is fundamental, but singular vectors v in the row space do more. (1) the v’s are orthogonal (2) the vectors $u=A v$ are also orthogonal (in the column space of A). Those v’s and u’s are columns in the singular value decomposition $A V=U \Sigma$. They are eigenvectors of $A^{T} A$ and $A A^{T}$, perfect for applications. We can list 10 reasons why orthogonal matrices like U and V are best for computation - and also for understanding. Fortunately the product of orthogonal matrices $V_{I} V_{2}$ is also an orthogonal matrix. As long as our measure of length is $\|v\|^{2}=v_{1}^{2}+\ldots+v_{n}^{2}$, orthogonal vectors and orthogonal matrices will win.