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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
- 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
- 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
- terminology - Must perpendicular (resp. orthogonal) lines meet . . .
In geometry books in French, there has traditionally been a very clear distinction between 'orthogonal' lines and 'perpendicular' lines in space Two lines are called perpendicular if they meet at a right angle Two lines are called orthogonal if they are parallel to lines that meet at a right angle Thus orthogonal lines could be skew (i e
- 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)
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