Usage of the word orthogonal outside of mathematics I always found the use of orthogonal outside of mathematics to confuse conversation You might imagine two orthogonal lines or topics intersecting perfecting and deriving meaning from that symbolize
Difference between Perpendicular, Orthogonal and Normal It seems to me that perpendicular, orthogonal and normal are all equivalent in two and three dimensions I'm curious as to which situations you would want to use one term over the other in two and
Are all eigenvectors, of any matrix, always orthogonal? In general, for any matrix, the eigenvectors are NOT always orthogonal But for a special type of matrix, symmetric matrix, the eigenvalues are always real and eigenvectors corresponding to distinct eigenvalues are always orthogonal
Eigenvectors of real symmetric matrices are orthogonal Now find an orthonormal basis for each eigenspace; since the eigenspaces are mutually orthogonal, these vectors together give an orthonormal subset of $\mathbb {R}^n$ Finally, since symmetric matrices are diagonalizable, this set will be a basis (just count dimensions) The result you want now follows
What is orthogonal transformation? - Mathematics Stack Exchange An orthogonal matrix can therefore be thought of as any "coordinate transformation" from your usual orthonormal basis $\ {\hat e_i\}$ to some new orthonormal basis $\ {\hat v_i\} $ You can view other matrices as "coordinate transformations" (if they're nondegenerate square matrices), but they will in general mess with your formula for the "dot