Currently I am studying orthogonal vectors and matrices. Although I got the theory, i.e. the connections between dot product, angle, orthonormal basis etc., I miss the intuition that lies behind certain formulas.
My question: How could you interpret it (geometrically) that the transpose of the orthogonal matrix A equals its inverse, viz. $A^{t} = A^{-1}$ or that for the determinant it is $det(A) = 1$.
Of course, one can prove all that analytically just fine. But how could I imagine the linear map and develop an intuition with respect to the 'right angle feature' of the vectors? I hope this is not too vague a question. If so, I will rephrase.
Best Answer
Think of 2x2 orthogonal matrix. If it's determinant 1, then it will be a matrix of the form $$ \begin{pmatrix} cos(\theta)&-sin(\theta)\\ sin(\theta)&cos(\theta)\\ \end{pmatrix} $$
So it transpose is $$ \begin{pmatrix} cos(\theta)&sin(\theta)\\ -sin(\theta)&cos(\theta)\\ \end{pmatrix} $$
First matrix is rotation by $\theta$ counter-clockwise and another is rotation by $\theta$ clockwise, so it makes sense they are the inverse of each other.
If it's determinant -1, think of it as a reflection matrix.
Higher dimensional matrices can be thought in a similar manner I believe.