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- 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 planes in n-dimensions - Mathematics Stack Exchange
To expound upon the definition of orthogonal spaces, you can prove that planes are orthogonal by using their basis elements Each (2d) plane has two basis elements and everything in the plane is a linear combination of them, so it suffices to show that both basis elements of one plane are orthogonal to both basis elements for another plane
- What is orthogonal transformation? - Mathematics Stack Exchange
Matrices represents linear transformation (when a basis is given) Orthogonal matrices represent transformations that preserves length of vectors and all angles between vectors, and all transformations that preserve length and angles are orthogonal Examples are rotations (about the origin) and reflections in some subspace
- 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
- What is the difference between diagonalization and orthogonal . . .
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- Difference between orthogonal and orthonormal matrices
The literature always refers to matrices with orthonormal columns as orthogonal, however I think that's not quite accurate Would a square matrix with orthogonal columns, but not orthonormal, change the norm of a vector?
- Orthogonal and symmetric Matrices - Mathematics Stack Exchange
Conversely, every diagonalizable matrix with eigenvalues contained in $\{+1,-1\}$ and orthogonal eigenspaces is of that form It follows that the set of your matrices is in bijection with the set of subspaces of $\mathbb C^n$
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