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- 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
- linear algebra - What is the difference between orthogonal and . . .
I am beginner to linear algebra I want to know detailed explanation of what is the difference between these two and geometrically how these two are interpreted?
- 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
- 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?
- How can three vectors be orthogonal to each other?
In this manner we end up with a description for an infinite family of orthogonal vectors, which hopefully makes it easy for you to convince yourself intuitively In a more general vector space, of course, this sort of pictorial intuition might no longer hold, but the idea of orthogonality can be easily generalised
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