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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 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 do we know that nullspace and row space of a matrix are orthogonal . . .
    How do we know that not only are these two subspaces orthogonal, they are also orthogonal complements because there is no vector in $\mathbb{R}^n$ that is perpendicular to the vectors in the nullspace that is not in the row space, and there is no vector perpendicular to the vectors in the row space that is not in the null space?
  • What is the orthogonal part? - Mathematics Stack Exchange
    The first explanation of orthogonal is that if you project b on C(A), the nature of moving the b toward C(A) based on the shortest path is where the orthogonality comes in But this seems more parallel to me than orthogonal? The second explanation is that the vector component of b orthogonal to C(A) is the orthogonal part of orthogonal projection
  • 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 $\endgroup$ –
  • 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





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