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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General solution using pseudo inverse.
I'm having trouble to understand the general solution of a $Ax=b$ when $x=A^+b+ [I-A^+A]w$
I don't understand why the $w$ is there and why $w$ can be any vector.
My view is:
$Ax=b$
$A[A^+b+ [I-A^+A]...
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2answers
45 views
Irreducible representations of $SO(2)$ on 2x2 matrices.
I'm having trouble verifying my understanding of the representation theory of Lie groups (which is minimal) with my experience playing around with the rotations of 2x2 matrices.
Specifically, if we ...
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1answer
50 views
A Question on the existence of an orthogonal matrix in Linear Algebra
The following is an exercise in my linear algebra textbook.
Suppose $\vec{x} \in \mathbb{R}^n$ and $\|\vec{x}\|^2 = 1$. Prove that there exists a matrix $A \in O(n,\mathbb{R})$ and $A^T=A$ such ...
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1answer
40 views
Identifying a “rotated shear” matrix
Suppose that we're in $\Bbb R^n$. Then the simplest shear matrix can be described as
$$S_\lambda =
\begin{bmatrix}
1 & \lambda & 0 & \ldots & 0 \\
0 & 1 & 0 & \ldots &...
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0answers
18 views
Parametrising a sparse orthogonal matrix [migrated]
I need to find a way to parametrise a matrix that is both sparse (to some degree) and orthogonal, i.e., I am looking for a parametrisation that describes $A \in \mathbb{R}^{n\times m}$ such that $A A^...
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2answers
41 views
Orthogonal matrices and matrix norms
I have seen some disagreement online and was wondering if anyone could clarify for me:
If $X$ is an arbitrary $n \times n$ matrix and $A$ is an arbitrary orthogonal $n \times n$ matrix, is it true ...
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2answers
35 views
find power of orthonormal matrix to get identity matrix
I have this orthonormal matrix:
$ Q =\frac{1}{9} \left(\begin{matrix}
4 & 8 & -1 \\
-4 & 1 & -8 \\
-7 & 4 & 4
\end{matrix}\right)$
If I calculate $Q^4$, I get the identity ...
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0answers
26 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 \\
...
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1answer
30 views
What's about the sum of two orthogonal vectors
I'd like to ask if the sum of two orthogonal vectors is a vector which is orthogonal on others, where can I get more details about that?
for example, suppose we have the Walsh matrix of 4, which is
...
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votes
1answer
74 views
Can an orthogonal non-symmetric 3x3 matrix have 3 real eigenvalues?
I was wondering if a non-symmetric orthogonal matrix can have his 3 eigenvalues in the real numbers.
All the 3 real eigenvalues orthogonal matrix i've found are symmetric.
Can someone give me a ...
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1answer
52 views
Conjugation of $\left(\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right)$
I'm interested in the following question.
Let $h=\begin{pmatrix} 0 & -1 \\ 1 & 0 \\ \end{pmatrix}$. This is an orthogonal map which is quite far away from the identity (say in the Frobenius ...
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1answer
34 views
(real symmetric matrices) does orthogonality of eigenvectors (distinct eigenvalues) depend on choice of basis?
For some reason I cannot wrap my head around this one. My schoolbook begins the chapter on eigendecomposition of real symmetric matrices by stating that eigenvectors from distinct eigenspaces are ...
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0answers
17 views
Invariant subspace SO(n)
Suppose $v\in\mathbb{C}$ is an eigenvector of $R\in SO(n)$ with nonreal eigenvalue $\lambda$. Let $V\subset \mathbb{R}^n$ be the two dimensional space spanned by $(v+\bar{v})/2$ and $(v+\bar{v})/(2i)$....
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1answer
44 views
Orthogonal diagonalization without eigenvectors
I stumbled onto a method for orthogonally diagonalizing a symmetric matrix with real entries and I was wondering what advantages (if any at all) it has over the eigenvector method.
It hinges on the ...
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1answer
27 views
Calculate matrix $A^T A$ with pairwise orthogonal vectors
I have a matrix $A$, that contains pairwise orthogonal vectors with length $1$, and I should calculate $A^T A$. I defined that:
$ v_{1}, v_{2}, v_{n-1}, v_{n} ∈ R^n \ and \ A ∈ R^{m _x n} $
and if I ...
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3answers
25 views
Trouble understanding orthonormal vectors.
I am pretty sure the definition of orthonormal matrix is a matrix whos columns contain vectors that are orthogonal to each other and all of length 1.
In that case I cant understand 17:04 of this ...
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1answer
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Induction step of proof of 'Every element of $O(n)$ is product of hyperplane reflections'
The following came up in induction step of proof of 'Every element of $O(n)$ is product of hyperplane reflections'.
An element $A$ in $O(n)$ is called hyperplane reflection if $$A=Pdiag(1,\cdots , 1,-...
6
votes
1answer
147 views
Can the derivative of the matrix absolute value explode when we approach singular matrices?
Let $ \text{GL}^+_n$ be the group of real $n \times n$ matrices with positive determinant, and consider the matrix absolute value function,
$| \cdot | : \text{GL}^+_n \to \text{Psym}$ given by $|A|=\...
1
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0answers
20 views
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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0answers
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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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0answers
28 views
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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0answers
31 views
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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votes
1answer
10 views
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 ...
1
vote
1answer
38 views
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 ...
4
votes
1answer
114 views
Eigenvalues of a real orthogonal matrix.
Let $A$ be a real orthogonal matrix. Then $A^{\text T} A = I.$ Let $\lambda \in \Bbb C$ be an eigenvalue of $A$ corresponding to the eigenvector $X \in \Bbb C^n.$ Then we have
$$\begin{align*}...
3
votes
1answer
49 views
How do I complete a matrix so that its columns are orthogonal?
I'm not sure how to start on these types of problems. I know that the dot product of the columns have to be zero between each pair.
Do I set up a system of linear equations for this? If so, what ...
0
votes
1answer
21 views
Transforming back and forth between reference frames using orthogonal transformation matrices
The transformation of a covariance matrix $C$ from reference frame 1 to reference frame 2 is described as
\begin{equation}
C_2 = R_{12}C_1R_{12}^T
\end{equation}
using the (orthogonal) ...
-2
votes
1answer
42 views
If matrix $A$ is unitary and $B^2 = A$ then $B$ is also unitary [duplicate]
I need to prove or give a counterexample:
If matrix $A$ is unitary and $B^2 = A$ then $B$ is also unitary
I think the statement is true since the unitary matrix A can only be Identity matrix I or ...
5
votes
1answer
110 views
Can we choose smoothly the singular vectors of a matrix?
Let $X$ be the space of all real $n \times n$ matrices, with strictly negative determinant, and pairwise distinct singular values. $X$ is an open subset of the space of all real square matrices. Is ...
0
votes
1answer
51 views
A “unique” solution to an equation over the orthogonal matrices?
Set $D=\text{diag}(-1,1,1,\dots ,1)$ be an $n \times n$ real diagonal matrix (where $D_{11}=-1$ and $D_{ii}=1$ for $i>1$).
Let $R,Q$ be special orthogonal matrices, satisfying $RDQ=D$. Is it ...
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vote
2answers
26 views
Show that $||v||^2 = ||P_0v||^2 + ||v - P_0v||^2$ for orthogoonal projection
I'm working on some practice problems from Noble & Daniel's Applied Linear Algebra (3rd), specifically here looking for help with question 5 from section 5.8 on pg. 232.
Suppose that $P_0$ is the ...
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1answer
47 views
Explain whether this matrix is symmetric or not?
I have a matrix $M$ and another $N$. $N$ is an orthogonal (orthogonal => $N^{T} = N^{-1})$ r x r matrix and $M$ is an r x r skew symmetric matrix (skew syemmtric => $M^{T} = -M$). Is $(N^{-1})$$(M^2)$$...
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1answer
26 views
The fundamental groups of 3-dimensional spherical space forms
Let $S^3/\Gamma_i\,(i=1,2)$ be a $3$-dimensional spherical space form, where $\Gamma_i \subset SO(4)$ is a finite subgroup acting freely on $S^3$. If $S^3/\Gamma_1$ is homotopy equivalent to $S^3/\...
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1answer
33 views
Are two isomorphic finite subgroups of $SO(4)$ conjugate?
Let $A,B$ be two finite subgroups of $SO(4)$ such that $A$ and $B$ are isomorphic as abstract groups. Can we find a $g \in SO(4)$ such that
$$
gAg^{-1}=B?
$$
If it is the case, does the same ...
0
votes
0answers
36 views
Expected value for the eigenvalue of random orthogonal matrix
I am interested in finding the expected value of the product of two eigenvalue with respected to the GOE ensemble.
So I have a random $n\times n$ matrix with all off diagonal elements $\mathcal{N}(0,...
2
votes
1answer
32 views
Homotopy groups of split orthogonal group
What are the homotopy groups of $O(d,d)$? Is it possible to compute them somwhow by using the fact that $O(d)\times O(d)$ is the maximal compact subgroup, and the homotopy groups of $O(d)$ are known?
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0answers
33 views
Are the invariants related in the characteristic equation of an orthogonal matrix (3x3 matrix)?
The characteristic equation of matrix A is
$$\lambda ^3 - I_1\lambda^2 + I_2\lambda-I_3 = 0 $$
For orthogonal matrix
$$I_3 = det(A) = \pm1$$
$$I_1 = tr(A)$$
Taking examples of orthogonal matrices, ...
0
votes
1answer
40 views
Problems with determining the orthonormal basis regarding an inner product $B$
Can somebody tell me where I made a mistake?
How would you approach such an exercise? Was my way too complicated?
$ B:\mathbb{R}^3\times \mathbb{R}^3\to \mathbb{R},\quad B(x,y)=\sum \limits_{i\neq j}^...
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votes
0answers
17 views
Counting $2\times2$ Orthogonal matrices over the ring $\Bbb{Z}_p[i]$. [duplicate]
Our research is about counting the number of orthogonal matrices over the ring of Gaussian integers modulo $p$.
A matrix $A$ is said to be orthogonal if $AA^T=I$.
My question is how many $2\times2$ ...
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1answer
44 views
For an orthogonal matrix $Q$, prove $\operatorname{cond}(Q)=1$
Given an orthogonal matrix $Q$, prove $$\|Q\|_2\cdot \|Q^{-1}\|_2=1$$
I succeed to solve it with eigenvalues but I'm looking for an easier way.
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2answers
68 views
Prove that special orthogonal matrix from $SO(n, \mathbb{R})$ has $\frac{(n-1)n}{2}$ independent parameters
Show that special orthogonal matrix from $SO(n, \mathbb{R})$ has $\frac{(n-1)n}{2}$ independent real parameters.
I assume that this will be related to Euler angles somehow or specifically to its ...
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0answers
33 views
What does the “standard basis” of $O(1,n)$ mean?
Let $O(1,n)$ be the orthogonal group of the quadratic form $b(x)=-x_0^2+\sum\limits_{i=1}^n x_i^2$. In other words, $$O(1,n)=\{T\in GL(n+1,\Bbb{R}|b(Tx)=b(x),\forall x\in R^{n+1}\}$$ Elements in $O(1,...
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2answers
45 views
Distance of matrix to $\mbox{SO}(n)$ w.r.t. Frobenius norm
Given $A \in \mathbb R^{n\times n}$, I was told that
$$\mbox{dist}(A, \mbox{SO}(n)) = \inf_{Q \in \mbox{SO}(n)} |A-Q| \overset ? = \inf_{Q \in \mbox{SO}(n)} | Q^\top A - \text{Id} |$$
where we use ...
2
votes
0answers
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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1answer
67 views
Orthogonal matrix whose first column is given
What would be an orthogonal matrix whose first column is $\underline{x} =
\begin{vmatrix}
-1\\
\underline{y}\\
\end{vmatrix}$,
where $\underline{y} \in {\rm I\!R}^{n-1} $, $\underline{x} \in {\rm I\...
0
votes
1answer
41 views
Matrix multiplication commutative property
Let A=(I-S)(I+S), can it be written as A=(I+S)(I-S) where I is identity matrix and S is n rowed real skew symmetric matrix? I have a question and in solution they wrote it. Thanks!
5
votes
2answers
80 views
Every real matrix $A$ is the linear combination of $4$ orthogonal matrices
Question:
Prove that every matrix $A\in M_n(\mathbb R)$ is the linear combination of $4$ orthogonal matrices $X, Y, Z, W$ , i.e. $A=aX+bY+cZ+dW$ for some $a,b,c,d\in\mathbb R$.
This problem is ...
0
votes
3answers
43 views
Question about a proof that the eigenvalues of an $n \times n$ orthogonal matrix are $\pm 1$
Suppose A: $n \times n$ orthogonal matrix,
$\lambda$ is an eigenvalue of A and $x$ is corresponding eigenvector.
We know that $Ax = \lambda x$
Then $(Ax)^T (Ax) = x^T A^T Ax = (Ax) \cdot (Ax) = |Ax|...
2
votes
1answer
52 views
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 ...
2
votes
2answers
43 views
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 = ...
