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
orthogonal vs orthonormal matrices - what are simplest possible . . . Sets of vectors are orthogonal or orthonormal There is no such thing as an orthonormal matrix An orthogonal matrix is a square matrix whose columns (or rows) form an orthonormal basis The terminology is unfortunate, but it is what it is
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 really is orthogonality? - Mathematics Stack Exchange If my reasoning is correct than, for any basis in a vector space there is an inner product such that the vectors of the basis are orthogonal If we think at vectors as oriented segments (in pure geometrical sense) this seems contradicts our intuition of what ''orthogonal'' means and also a geometric definition of orthogonality
How to find an orthogonal vector given two vectors? Ok So taking the cross product gives me orthogonal vector in $\mathbb {R}^3$ And how to approach the same question in $\mathbb {R}^2$ for example I mean with two vectors each having two componetns?