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.

I recently took linear algebra course, all that I learned about orthogonal matrix is that Q transposed is Q inverse, and therefore it has a nice computational property. Recently, to my surprise, I learned that transformations by orthogonal …

The original question was asking about a matrix H and a matrix A, so presumably we are talking about the operator norm.

This is because SVD works by finding a right and a left orthogonal (rotation) matrix, which rotates the matrix in question to column resp row orthogonality. But the nxn orthogonal matrix is already row and column-orthogonal. Thus the SVD routine has no rotation-criterion.

2015年9月1日 · I've seen the statement "The matrix product of two orthogonal matrices is another orthogonal matrix. " on Wolfram's website but haven't seen any proof online as to why this is true. By orthogonal m...

2020年6月3日 · A unitary matrix is just the complex number version of an orthogonal matrix. So if the matrix is real then it is orthogonal if and only if it is unitary. I do prefer the word unitary, because it reminds what you need to check - the columns (or rows) are perpendicular to each other and each row/column has unit length (length 1).

2015年5月2日 · The term "orthogonal matrix" probably comes from the fact that such a transformation preserves orthogonality of vectors (but note that this property does not completely define the orthogonal transformations; you additionally need that the length is not changed either; that is, an orthonormal basis is mapped to another orthonormal basis).

2016年4月11日 · An orthogonal matrix in $\mathbb{R}^{3\times3}$ with real eigenvalues is diagonalizable. 0.

2017年4月3日 · Nearest semi-orthogonal matrix using the entry-wise $ {\ell}_{1} $ norm 3 For a positive definite symmetric matrix, the orthogonal diagonalization is a SVD of A.

2012年5月22日 · $\begingroup$ The columns of an orthogonal matrix are also assumed to have norm one, so an orthogonal matrix is actually "orthonormal" from this perspective. $\endgroup$ – t.b. Commented May 22, 2012 at 13:49