Usage of the word orthogonal outside of mathematics In debate(?), "orthogonal" to mean "not relevant" or "unrelated" also comes from the above meaning If issues X and Y are "orthogonal", then X has no bearing on Y If you think of X and Y as vectors, then X has no component in the direction of Y: in other words, it is orthogonal in the mathematical sense
linear algebra - What is the difference between orthogonal and . . . Two vectors are orthogonal if their inner product is zero In other words $\langle u,v\rangle =0$ They are orthonormal if they are orthogonal, and additionally each vector has norm $1$ In other words $\langle u,v \rangle =0$ and $\langle u,u\rangle = \langle v,v\rangle =1$ Example For vectors in $\mathbb{R}^3$ let
orthogonal vs orthonormal matrices - what are simplest possible . . . Generally, those matrices that are both orthogonal and have determinant $1$ are referred to as special orthogonal matrices or rotation matrices If I read "orthonormal matrix" somewhere, I would assume it meant the same thing as orthogonal matrix Some examples: $\begin{pmatrix} 1 1 \\ 0 1 \end{pmatrix}$ is not orthogonal
How to find the orthogonal complement of a given subspace? In this case that means it will be one dimensional So all you need to do is find a (nonzero) vector orthogonal to [1,3,0] and [2,1,4], which I trust you know how to do, and then you can describe the orthogonal complement using this
linear algebra - Unitary transformation and Orthogonal transformation . . . The orthogonal matrices themselves are important in Lie group theory Unitary matrices don't preserve dot product, they preserve a Hermitian form, which is not bilinear but is instead conjugate linear in the second argument, and it is positive definite in the same sense as the dot product
How do you orthogonally diagonalize the matrix? $\begingroup$ The same way you orthogonally diagonalize any symmetric matrix: you find the eigenvalues, you find an orthonormal basis for each eigenspace, you use the vectors in the orthogonal bases as columns in the diagonalizing matrix $\endgroup$ –
linear algebra - Why is an orthogonal matrix called orthogonal . . . $\begingroup$ @Freeze_S : If you're only talking about matrices, then orthogonal maps coordinate lines in one orthogonal frame to orthogonal lines in another Unitary does the same thing, but in complex spaces For smooth coordinate systems, orthogonal is a little different