- 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
- 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 If the eigenvalues are not distinct, an orthogonal basis for this eigenspace can be chosen using Gram-Schmidt
- 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)
- If $X$ is an orthogonal matrix, why does $X^TX = I$?
$\begingroup$ @julien Since the answer is asked the way it is, I take it his definition is: an orthogonal matrix is a square matrix whose column vectors constitute an orthonormal set of vectors (or orthonormal basis) $\endgroup$ –
- What is the difference between diagonalization and orthogonal . . .
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- Eigenvalues in orthogonal matrices - Mathematics Stack Exchange
An orthogonal transformation is either a rotation or a reflection I will focus on 3D which has lots of
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