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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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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
48 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
32 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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15 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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40 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
24 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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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
36 views

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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23 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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29 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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1answer
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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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1answer
36 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 ...
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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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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) ...
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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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1answer
109 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 ...
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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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2answers
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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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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
24 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
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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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33 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,...
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1answer
30 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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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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1answer
43 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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2answers
39 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 ...
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64 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
61 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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1answer
40 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!
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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 ...
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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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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 = ...
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19 views

Householder matrices for thin QR updating

Suppose you have a QR decomposition of the form: $X=QR$ (Where $X$ is an arbitrary matrix of size $(n \times p)$, $Q$ is an orthogonal matrix of size $(n \times n)$ and $R$ is an upper trapezoidal ...
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2answers
107 views

Does 3x4 matrix have an Inverse? Why? [duplicate]

I saw this question somewhere and made me think do 3x4 matrices have an inverse, as I previously that that only square matrices have an inverse. If non-square matrices have an inverse, especially if ...
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31 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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50 views

Orthogonal matrix multiplied by diagonal matrix multiplied by transpose of the orthogonal matrix

Suppose tall matrix $A$ is $n \times k$ and that its columns are orthogonal, i.e., $A' A = I_k$. Suppose further that diagonal $M$ is $n \times n$ and has either $1$ or $0$ on its main diagonal. ...
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36 views

Proof based on orthogonal matrix

I'm hoping to show that $|Q| = +1$ or $-1$ if $Q$ is a $p \times p$ orthogonal matrix. Since I know that $|QQ'| = |I|$ and $|Q||Q'| = |Q|^2$, then $|Q|^2 = |I|$ How should I approach this proof(...
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0answers
12 views

Why is the maximum value of an orthonormal dictionary of size $N \times N$ less than that of another having size $M \times M$, where M>N?

I was trying to compare the maximum value in DCT dictionary (representing an orthonormal dictionary) of size $N \times N$ to that of another DCT dictionary of size $M \times M$, where $M>N$. I ...
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2answers
39 views

Reducing the quadratic form

I'm trying to reduce the quadratic form $q(x_1, x_2, x_3, x_4) = x_1x_2 + x_1x_3 + x_1x_4 + x_2x_4$ into a quadratic form of the form $q = λ_1y_1^2 + λ_2y_2^2 + ··· + λ_ry_r^2$ for some real numbers $...
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3answers
64 views

When does $P\text{SO}P^{-1} \subseteq \text{SO}$?

Let $P \in \text{GL}_n^{+}(\mathbb {R})$. Suppose that $P\cdot \text{SO}(n)\cdot P^{-1} \subseteq \text{SO}(n)$. Is it true that $P \in \lambda \text{SO}(n)$ for some $\lambda \in \mathbb{R}$? I ...
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0answers
16 views

Index notation - notation for the inverse change of basis matrix with a hermitian metric

So, when we have any symmetic bilinear form $g = g_{ij} \epsilon^i \otimes \epsilon^j$, we can write $(A^{-1})^\mu{}_i = A_{\space i}{}^\mu$. This is one of the most beautiful things that index ...