Questions tagged [orthogonal-matrices]

Matrices with orthonormalized rows and columns. An orthogonal matrix is an invertible real matrix whose inverse is equal to its transpose. For complex matrices the analogous term is *unitary*, meaning the inverse is equal to its conjugate transpose.

453 questions
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Are the derivatives of the orthogonal polar factor locally integrable?

Let $\mathbb{D}^2$ be the closed unit disk, and let $f:\mathbb{D}^2 \to \mathbb{R}^2$ be a real-analytic map, satisfying $\det df>0$ everywhere except on a set of Hausdorff dimension not greater ...
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Generate orthogonal (lower) upper-triangular matrices

Problem I am trying to numerically verify the fact that "the orthogonal lower (upper) triangular matrix has to be diagonal". However, I have difficulty finding general matrices that satisfies both ...
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Then which of the following statements are true?[CSIR-2018-December]

Let $\{u_1,u_2,..., u_n\}$ be an orthonormal basis of $\mathbb {C^n}$ as column vectors. Let $M=(u_1,u_2,...,u_{k})$ and $N=(u_{k+1},u_{k+2},...,u_{n})$ and $P$ be a $k \times k$ diagonal matrix with ...
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Find a unitary basis of the $\mathbb{R}$-vector space of $n \times n$ (complex) Hermitian matrices

The question is on the title ($n \in \mathbb{N}^*$). To be clear, unitary basis here means basis consisting of (complex) unitary matrices. I wonder this question because recently I've read about ...
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Is relationship of orthonormality & orthogonality equivalent to that of squares & rectangles?

To confirm my analogy, I am asking if I can consider every orthonormal basis to be an orthogonal one (but not vv) in the same sense that every square is a rectangle (but not vv). I believe the answer ...
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Can $\mathfrak{u}(n)$ be decomposed as direct sum of the sets of symmetric and skew-symmetric real matrices?

It is a well-known result (proved for example also in this answer) that $\mathfrak{gl}(n,\mathbb C)\simeq \mathfrak{u}(n)_{\mathbb C}$, which can also be understood as another way to state that any ...
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Complex-analytic isometry of hyperbolic ball and half-space

I'm trying to prove that the $n$-dimensional Poincaré ball and half-space models of hyperbolic space are isometric. Here the Poincaré ball $\mathbb B^n_R$ ($n$-ball of radius $R$) has the Riemannian ...
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Haar measure from axis-angle representation of $SO(3)$
My aim is to study some special representations of $SO(3)$ using characters, and to do this I need an explicit expression of the Haar measure on $SO(3)$. I've found some "versions" of the Haar measure ...
Matrices in $\operatorname{O}(n) \setminus \operatorname{SO}(n)$
Can any matrix $M \in \operatorname{O}(n) \setminus \operatorname{SO}(n)$ be written as $I_n - uu^T$ where $I_n \in \mathbb{R}^{n \times n}$ is the identity matrix and $u \in \mathbb{R}^n$, \$||u||_2 = ...