What's the insight for a 3x3 matrix with orthogonormal columns,the rows are also orthogonormal? I know this can be easily proved with simple matrix tricks,
But I don't know the insight for this, and just feels it amazing that if I pick up 3 orthogonormal vectors in 3d space, their corresponding x,y,z portions automatically forms orthogonormal basis,too!
I've been googling a lot with no satisfactory answer, hope I can find it here,
Thanks.

A square  real matrix $M$  has columns ON if and only if $M^{-1}=^tM$ if and only if $(^tM)^{-1}=M=^t(^tM)$  if  and  only  if  $^tM$  columns are ON if and only if $M$ lines are ON.
ON = orthonormal. 
