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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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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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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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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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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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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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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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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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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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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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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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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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(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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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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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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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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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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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,-...
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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|=\...
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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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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*}...
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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 ...
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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) ...
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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 ...
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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 ...
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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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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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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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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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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 ...
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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,...
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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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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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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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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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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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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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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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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 ...
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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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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\...
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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!
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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 ...
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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|...
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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 ...
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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 = ...