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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.

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Geometry of $Sp(2N,R)/U(N)$

Given $\mathbb{R}^{2N}$ equipped with a symplectic form $\Omega^{ab}$ and a compatible symmetric, positive definite, bilinear form $G^{ab}$, we can look at the symplectic group $\mathrm{Sp}(2N,\mathbb{...
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Is $Av_1,Av_2,Av_3$ orthogonal if you have eigenvector of $A^TA$

Let $A\in M_3(\mathbb R)$ and if $v_1,v_2,v_3$ orthonormed eigenvectors of matrix $A^TA$ and which eigenvalues is $1,2,3$ then vectors $Av_1,Av_2,Av_3$ is orthogonal? I only know that we need to ...
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324 views

How many independent parameters are in a skew-symmetric as well as orthogonal matrix?

I'm currently trying to parameterize a given real and square matrix $A$ with the properties $A^T=-A$ and $A^TA=\textbf{1}_N$, for even $N$. I don't know how many independent parameters I would have ...
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216 views

Singular value decomposition of a low rank weak diagonally dominant M-matrix. When is the unitary polar matrix positive semi-definite?

Let $A$ be an $n \times n$, non-symmetric, real, weak diagonally dominant M-Matrix. Its diagonal is strictly positive, its off-diagonal is negative or zero and all its columns sum to zero. $A$ has ...
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66 views

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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Higher dimensional version of “a product of two reflections is a rotation”

In higher dimensions, which orthogonal matrices are a product of two reflections? Is it all of $SO(n)$? In the complex case is it $U(n)$?
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Name of orthogonal/unitary matrix decomposition?

Suppose we have an orthogonal/unitary matrix $T$ of even dimension. Then we can decompose it into: $$ T = \begin{bmatrix} U_1 & 0 \\ 0 & U_2 \end{bmatrix} \begin{bmatrix} R & -\sqrt{...
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Instability of even windings of SO(4)

Representatives of $\pi_1(SO(2))=\mathbb{Z}$ may be given by paths$$\theta\mapsto\left(\begin{array}{lr}\cos(n\theta)&-\sin(n\theta)\\\sin(n\theta)&\cos(n\theta)\end{array}\right).$$However, $\...
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129 views

Rotationally Invariance Of Probability Density Functions

If $X_1,X_2$ are independent, normally distributed (with common variance) random variables, then the probability density function of the random vector $(X_1,X_2)$ in ${\bf R}^2$ is rotationally ...
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Questions on change of coordinates with respect to orthonormal bases in $\;\mathbb R^2\;$

Let $\;U \subset \mathbb R^2\;$ be a smooth open domain. On $\;\partial U\;$ consider the positively oriented orthonormal basis $\;(\mathcal v,\mathcal τ)\;$ where $\;\mathcal v\;$ is the outer ...
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Show that Gram-Schmidt$(Y)$ = $B$Gram-Schmidt$(X)$ such that $B$ is an orthogonal matrix

Let $\begin{align} V_{k,n} &= \Bigl \{ \begin{pmatrix} v_{1} \\ v_{2} \\ \vdots \\ v_{k} \end{pmatrix} : v_i\in \mathbb{R}^n\text{ and } \{v_1,\...
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Homotopy between $O(p,q)$ and $O(p)\times O(q)$

Suppose that $p,q$ are two positive integers. Let $O(p,q)$ be the set of all the $(p+q)\times(p+q)$ real matrices $A$ satisfying $A^Tdiag(I_p,-I_q)A=diag(I_p,-I_q)$. Let $O(n)$ be the set of all the $...
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Elements of the indefinite orthogonal group that leave invariant a positive definite symmetric bilinear form

new to StackExchange, please correct me if I do something wrong.. Let $X$ be a matrix of the indefinite orthogonal group $\mathrm{O}(p,q)$, that is to say $X^T I_{p,q} X = I_{p,q}$ where $I_{p,q}$ ...
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Set invariant under group action

I am reading a paper with the following description: $O(n): \{Y\in \mathbf{R}^{n\times n}\mid Y^TY=I\}$ We say a set $V$ is $T$-invariant if $TV\subseteq V$, where $T$ is a linear transform. So ...
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Norms (eigenvalues) of sums of orthogonal matrices

Let $T_1, \ldots, T_n$ be a set of real-valued symmetric matrices satisfying $Tr(T_j T_k) = 0$ for all $j\neq k$. Consider the norm $\|T\|_{\infty} = \max_{\|M\|_1 \leq 1} \operatorname{Tr}\left[M^T ...
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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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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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Are the invariants related in the characteristic equation of an orthogonal matrix (3x3 matrix)?

The characteristic equation of matrix A is $$\lambda ^3 - I_1\lambda^2 + I_2\lambda-I_3 = 0 $$ For orthogonal matrix $$I_3 = det(A) = \pm1$$ $$I_1 = tr(A)$$ Taking examples of orthogonal matrices, ...
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What does the “standard basis” of $O(1,n)$ mean?

Let $O(1,n)$ be the orthogonal group of the quadratic form $b(x)=-x_0^2+\sum\limits_{i=1}^n x_i^2$. In other words, $$O(1,n)=\{T\in GL(n+1,\Bbb{R}|b(Tx)=b(x),\forall x\in R^{n+1}\}$$ Elements in $O(1,...
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32 views

Minimum singular value of sum of rotations

Consider orthogonal matrices $Q_1, Q_2 \in \mathbb{R}^{d \times d}$, where both matrices are proper rotations of angle $\theta_1, \theta_2$ around different axes. Now, consider the symmetric matrix $$...
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How to show that $SO(3)$ and $\mathbb{R}P^3$ are dffeomorphic?

Pardon me for repeating a question, but I am not able to justify one aspect of this proof that $SO(3)$ and $\mathbb{R}P^3$ are diffeomorphic, and I was hoping that someone could clarify it for me. ...
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Equivalence classes of orthogonal matrices

Consider $O(n)$, the Lie group of $n{\times}n$ orthogonal matrices. Let $\pi$ be a given permutation of $(1,2,\ldots,n)$; the elements of the corresponding permutation matrix are \begin{eqnarray} P_{...
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37 views

On a special decomposition of a $3\times 3$ matrix

Let $A\in\mathbb{R}^{3\times 3}$ be a diagonalizable matrix with strictly positive eigenvalues. (Note that $A$ is not required to be symmetric.) Let $A_S$ be the symmetric part of $A$, that is $$ A_S ...
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42 views

$SO(N)$ generators to generate a basis for the space of $N\times N$ matrices

The generators of $SO(N)$ can be written as $(L_{ab})_{ij}=\delta_{ia}\delta_{jb}-\delta_{ja}\delta_{ib}$, with $1\leq i,j\leq N$ and $1\leq a<b\leq N$. Obviously, these generators for a basis for ...
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24 views

Transformation matrix for Higgs Doublet

If we consider two complex Higgs doublets with $SU(3)_{c} ⊗ SU(2)_L ⊗ U(1)_Y$ quantum numbers $\phi_{A,B} \sim (1,2,1)$, such that $$\begin{align}&\phi_A = \begin{pmatrix} \phi_{A}^{\dagger}\\ \...
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131 views

Quaternion Converted to Rotation Matrix then Derived with Respect to this Quaternion

I was wondering how a derivative of a rotation matrix generated based on a quaternion and then differentiated with respect to this quaternion would be calculated. $$q_{1\times 4} = [q_0 q_1 q_2 q_3]^...
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Does the following unitary matrix factorization have a name?

I know any unitary matrix can be factored as follows: $$\underline {\overline {\bf{U}} } = \left( {\prod\limits_{j = N}^1 {\underline {\overline {\bf{\Psi }} } \left( {{{\underline {\bf{w}} }_j}} \...
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Orthogonal Procrustes Problem Proof

I'm going through the proof for the Orthogonal Procrustes problem and I was able to derive all but the last few steps. Namely, the implication from lines $6 \Rightarrow7 \Rightarrow 8$ (I was able to ...
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$L_{1,1}$-norm and orthonormal matrices

Given any point $p \in \mathbb{R}^n$ and two matrices whose columns are orthonormal $X \in \mathbb{R}^{n \times j}$, $Y \in \mathbb{R}^{n \times (n - j)}$, such that $Y$ spans the orthogonal ...
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Isotropic tensor field depending a vector

I am wondering how to prove the following statement (which is widely used, for example in turbulence theory) mathematically rigorously: Assume we are talking about $V=\mathbb{R}^3$. Given a tensor ...
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How to orthogonally project to boundary of the ball $\mathcal{B}_r(0)\triangleq\{Y\in\mathbb{R}^{n\times n}:\|Y\|_2<r\} $?

Let $r>0$, and let $A\notin\mathcal{B}_r(0)\triangleq\{Y\in\mathbb{R}^{n\times n}:\|Y\|_2\leq r\}$, where $\|\cdot\|_2$ is the induced 2-norm. Let $\bar A$ be the orthogoanl projection of $A$ on ...
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Prove that a real $2 \times 2$ matrix is orthogonal if and only if it is of one of the following forms: Proof Review And Possible Author Error?

Prove that a real $2 \times 2$ matrix is orthogonal if and only if it is of one of the forms $\begin{bmatrix} a & b \\ -b & a \end{bmatrix}$, $\begin{bmatrix} a & b \\ ...
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Show that the matrix $P=I-2hh^T$ is orthogonal and find its first column.

Let $x=(x_1,...,x_n)^T$ a column vector in $\mathbb{R}^n$ so that $x_1\neq -1.$ Let $h$ a unitary vector in the direction of $x-e_1$ where $e_1$ is the vector in $\mathbb{R}^n$, $e_1=(1,0,...,0)$. ...
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311 views

Show when the inequality for matrix-vector multiplication for the 2 norm is an equality?

I am asked to show for which vectors the inequality $||Ax|| \le ||x||||A|| $ is an equality. My intuition tells me that this happens when $x$ is in the direction of the right singular vector ...
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108 views

Circulant Orthogonal $\operatorname{MDS}$ Matrix

Definition: A matrix $M$ of order $n$ over a field is a $\operatorname{MDS}$ matrix if and only if every sub-matrix of $M$ is non-singular. My question: How to proof the following statement. If $A$ ...
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28 views

When is there an orthogonal matrix that does this transformation.

Let $x,y,z,t\in [0,1]$ such that $x^2+y^2=z^2+t^2=1$ and $\alpha,\omega \in [0,\frac{\pi}{2}]$ be given parameters. I'm trying to find out when there is an orthogonal (real) $4\times 4$ matrix $$P = \...
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Restricted unitary transformations on separable vectors

Recently, I've been interested in the restriction of operators on complex Hilbert spaces to the real numbers. This seems like an interesting way to quantify how "complex" a vector is with respect to ...
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How to plot $SO(3)$ to unit sphere?

I understand that an $SO(3)$ elements corresponds to a $3 \times 3$ rotation matrix, and it could be mapped to $\omega \in \mathfrak{so}(3)$ (lie algebra), quaternion and Euler angles. But how to ...
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Move an orthonormal frame towards principal axis of a p.s.d. matrix

Be $\mathcal{W}=\{W=(w_1|\dots|w_p)\in\mathbb{R}^{n\times p}\;:\;W^TW = I\}$ the set of $p$-dimensional orthonormal frames in $\mathbb{R}^n$. Consider $$ L(W) = \mathrm{tr}(MP_W) = \mathrm{tr}(MWW^T) ...
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151 views

Orthogonal columns of a nonsquare matrix

Consider a SVD of a $3 \times 2$ matrix $A$, why the product of a $2 \times 2$ $S$ orthogonal matrix $AS$ has orthogonal columns while the product of a $3 \times 3$ orthogonal matrix $SA$ won't? I ...
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None element of orthogonal matrix can't have unit modulus larger then 1

None element of orthogonal matrix can't have unit modulus larger then 1. I've tried to use the properties of orthogonal matrices ( $|det(A)| = 1$ and $Q^T=Q^{-1}$ ) but I couldn't find out how they ...
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209 views

Uniformly random matrices in SO(n)

I don't know whether this question fits best on StackOverflow, Math.SE or CrossValidated; I am looking for a practical algorithm to generate uniformly distributed matrices in $SO(n,\mathbb{R})$ (aka ...
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86 views

Semidirect product structure on orthogonal group

Using that the orthogonal group $O(2n)$ consists of two connected components, which are $SO(2n)$ and $RSO(2n)$ (where $R=\operatorname{diag}(-1,1\ldots,1)$), one can construct a diffeomorphism $\phi:O(...
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Orthogonal Block Matrix, the relation between the block Matrices.

I am trying to find a relation between $U$ and $V$, such that U is a $(p\times d)$ semi-orthogonal matrix (i.e $\ U^{T}U=\mathbf{I}_{d}$) and $V$ is a $(p\times (p-d))$ matrix which is the completion ...
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51 views

Normal Subgroup of O(1,1)

Let $$O(1,1)=\{A\in GL_n(F):A^tJA=J\}$$ where $$J= \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}$$ My question is; Let $$T=\{A\in O(1,1): A=\begin{pmatrix} ...
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241 views

similar transformation of rotation matrix by orthogonal matrix

I am reading an old paper by Horn (see ch.4): https://www.jstor.org/stable/2372705?seq=1#page_scan_tab_contents And for learning convenience of users, the following is a similar discussing: ...
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23 views

Is it true $GS(TA)=T GS(A)$ where GS denotes the Gram Schmidt process and $T \in O(n,\mathbb R)$?

Let $A$ denote an $n \times n$ matrix.Suppose $GS:GL(n,\mathbb R) \to O(n,\mathbb R)$ denotes the Gram Schmidt orthogonalization process.Is it true that GS(TA)= TGS(A) where $T$ is an orthogonal ...
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32 views

Polar of orthogonal set invariant under group action

I just ask the following question: Set invariant under group action Furthermore, How to prove the green part Original paper: http://arxiv.org/pdf/1403.4914v1.pdf (p.1324) Let $$O(n)=\{...
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109 views

Proof of the rotation matrix is an extreme point of $\text{conv } SO(n)$

Define the set of rotation matrices: \begin{equation} \begin{aligned} SO(n) := \{X\in \textbf{R}^{n\times n}: X^TX=I, \text{det}(X)=1\} \end{aligned} \end{equation} I want to prove that if $X\in SO(...
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25 views

Translation as product of reflections

I am facing the following problem, given the translation in the euclidean affine space of dimension 4 $ \tau_v= \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ ...