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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Converting a linear space into a orthonormal base

I've been given a linear space made by 2 equations $U = \{\ {(x,y,z,w) ∈ R^4: x - y + z = 0 \land y - z + w = 0} \}\ $ and I have to get the orthonormal base of this equation and what I did to to that ...
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Distance of a vector from a subspace made by equations in Linear Algebra

So I have an exercise which gives my me this subspace : $U = {(x,y,z) ∈ R^3: x - y - z = 0 \text{ and } x + y + z = 0}$ and i have to determine the orthogonal projection of $(3,0,-1)$ which i did, ...
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Question about SVD and orthogonal matrices

Let $X$ be a $m \times n$ matrix. By SVD, I obtain $X = UDV^T$, where $U$ and $V$ are both orthogonal matrices, and $D$ is a diagonal matrix. I think the following is true (but not sure why): $(VDV^T +...
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Number of elements in a set of orthogonal matrices

Let $S$ be defined as $$S:= \left\{ M \in \mathbb{F}_3^{2 \times 2} : M \text{ is orthogonal} \right\}$$ where $\mathbb{F}_3$ is a field with $\mathbb{F}_3 = \{ 0, 1, 2\}$. How many elements does the ...
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Can these matrices be multiplied in $\mathcal O(n^2)$ time?

Consider a real-valued orthogonal matrix $Q$ and a sequence of diagonal matrices $\{D_m\}_{m=1}^\infty$. All entries of $Q$ are real and the entries of each $D_n$ are real and positive. What is the ...
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Product of the complex eigenvalues of real orthogonal matrices is non-negative

How to prove that Product of the complex eigenvalues of real orthogonal matrices is non-negative? My attempt: If $Ax=\lambda_1 x$ and $Ay=\lambda_2 y$ then $$\langle x,y\rangle=\langle AA^tx,y\...
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How to solve this orthogonal matrix equation?

I have an equation like this: $ A = R^T * B * R $. Actually, the problem should be like this: $ argmin\sum_{i=0}^{n-1}(q_i^T*R^T*B*R*p_i)$. It's similar to the above equation. $q_i, p_i$ are known. $ ...
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Under what conditions can I orthogonally diagonalize a matrix

I know that you can orthogonally diagonalize a matrix if it's symmetric. Under what other conditions can I orthogonally diagonalize a matrix? And if a matrix is diagonalizable, is it orthogonally ...
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Does spinor norm of the restiction to some certain subspace of an orthogonal transformation with spinor norm $+1$ change?

Let $V$ be a vector space over a finite filed $F_q$ with dimension $n$, where $q$ is a power of some odd prime $p$ and $n=2m$ with $m$ odd. Let $x\in {\rm SO}_n^\epsilon(q)$ and $x^2=-1_V$, then $V$ ...
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Eigenvectors of $A \in SO(2n)$ and $A \in SO(2n+1)$

Does every matrix $A \in SO(2n)$ have an eigenvector? Does every matrix $A \in SO(2n+1)$ have an eigenvector? I think that you can answer both questions with yes, is that true?
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If $A$ is an orthogonal matrix with $|A|=-1$, show that $|I-A|=0$

Let $A$ be an $n \times n$ orthogonal matrix where $A$ is of even order with $|A|=-1.$ Show that, $|I-A|=0,$ where $I$ denotes the $n \times n$ identity matrix. My approach $A \cdot A^{\top}=I$ $|A| \...
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Can an orthogonal matrix move monotonically toward a signed permutation matrix?

The question is motivated by this question on Mathematics SE. Let $A \in O(n)$ be an orthogonal matrix that is not a signed permutation matrix, and let $P$ be the nearest signed permutation matrix to $...
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skew-diagonalizing an anti-symmetrc matrix

Let's assume that i have a (real) $2N\times2N$ anti-symmetric matrix $B=\left\{ b_{ij}\right\} $ with the property that $BB^{T}=\boldsymbol{1}$ where $\boldsymbol{1}$ is the identity matrix. Is it ...
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Product of two symmetric matrices: If one orthogonal matrix is known, find the other one

Let $T \in \mathbb{R}^{n \times n}$ be a product of two symmetric matrices $A, B \in \mathbb{R}^{n \times n}$: $$T = AB$$ By the Spectral Theorem, $A$ and $B$ each have an eigendecomposition $A = ...
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Orthogonal matrices $A$ for which $A^n = I$

Can we say that orthogonal matrices for which $P^n = I$ are necessarily permutation matrices? I know the matrices that satisfy the condition $P^n = I$ are called periodic. Also, permutation matrices ...
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Proof that greatest entry of a unit vector is $\leq 1$ and $\geq \frac{1}{\sqrt{n}}$

I have a real orthogonal matrix so the column vectors form an orthogonal system and thus the vectors have length one. I now want to show that for an arbitrary column vector $v_k \in \mathbb{R^n}$ the ...
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Orthogonal complements really confuse me, I think its the notation?

For example what do I do here, I know wha to do for part a but then...? Let$$W=\operatorname{Span}\left\{\left(\begin{array}{c}1\\1\\0\\0\end{array}\right),\,\left(\begin{array}{c}0\\1\\1\\-1\end{...
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Orthogonal Vector Space with Identity Matrix and Orthogonal Projection

I'm doing some homework and I'm having some trouble trying to prove something about orthogonal vector spaces. Given a vector space $S \subseteq \mathbb{R}^m$ and $P$ an orthogonal projection on $S$ it ...
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Proving determinant for orthogonal matrices

If there are two matrices $\det(A)$ and $\det(B)$ such that $\det(A)+det(B)=0$, where both the matrices are real orthogonal matrices. How can I say the following? $\det(A+B) = \det(A^T(A+B)B^T)$ Is it ...
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Vector field such that outer product is equivariant to orthogonal transformations and change of basis. Does it exist?

I am searching for a compactly supported, continous, non-zero function $G:\mathbb{R}^{d}\to\mathbb{R}^{d}$ satisfying \begin{align*} MG(x)G(x)^{T}M^{T}=G(Mx)G(Mx)^{T} \end{align*} for all orthogonal ...
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Neighborhood in orthogonal group

Let $A\in O(n)$. Assume that $|a_{i,i}|\neq 1$ for every $i$. Prove that in every neighborhood of $A$ there exists $B\in O(n)$ such that $|b_{i,i}|>|a_{i,i}| \text{ for every } i \text{ and } |b_{i,...
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Why is every element in $SO(3)$ is a rotation?

Question: I am worried about the proof of Equivalence of an orthogonal matrix to a rotation matrix provided in this Wikipedia page, for the change of basis matrix may be a complex but not real matrix, ...
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General form of an orthogonal matrix involving angles, sines, and cosines

In $\mathbb{R}^2$, all orthogonal matrices are one of two forms: $$\begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix} \text{ or } \begin{pmatrix} \cos\theta &...
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$\forall n \geq 3, \text{Span}(SO_n(\mathbb{R})) = M_n(\mathbb{R})$

I would like to prove that the following holds for all $n \geq 3$ : $$\forall n \geq 3, \text{Span}(SO_n(\mathbb{R})) = M_n(\mathbb{R})$$ So far I have noticed the following : For $n =2$ the ...
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Isomorphism between $SO_2\tilde{\times}\mathbb{Z}_2$ and $O_2$

This is the exercise 23.10 p. 135 of Groups and symmetry of Armstrong : Let $G$ be an abelian group and write $G \tilde{\times}\mathbb{Z}_2 $ for the semidirect product $G\rtimes_\phi\mathbb{Z}_2$, ...
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Are orthogonal operators always isomorphisms?

I need to show the following: Let $V$ be a finite dimensional vector space with inner product, if $T$ is orthogonal, show that T is injective and surjective I think it is injective because T ...
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unitary operator explanation [closed]

the unitary operator definition is $ ⟨Tv,Tw⟩ = ⟨v, w⟩$ for every $v, w$ in $V.$ can you please explain the intuition and what the formal definition actually means? why unitary operator preserves the ...
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Solving system of linear equations using orthogonal matrix

I'm given the following matrix: $$ A = \begin{pmatrix} \frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}} \\ 0 & 1 & 0 \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \end{pmatrix} ...
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How do you call a combination of rotation and uniform scaling?

Is there established standard terminology for a linear transformation $T$ that is a combination of a rotation and uniform scaling, that is, it can be written as $T=\alpha O$ with a scalar $\alpha\in\...
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Othorgonal matrices and full rank

I have a question regarding orthogonal matrices. Is it the case that orthogonal matrices always have full rank? I tried to illustrate a $2\times 2$ orthogonal matrix with $\det=-1$ and come to the ...
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If $V$ is right-orthogonal, does it hold $\langle AV,BV\rangle_F=\langle A,B\rangle_F$?

Let $A,B\in\mathbb R^{m\times n}$. It's easy to see that for the Frobenius inner product it holds $$\langle A,B\rangle_F=\operatorname{tr}B^\ast A=\operatorname{tr}A^\ast B.\tag1$$ So, if $U\in\mathbb ...
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determinant of an orthogonal matrix

The question goes like this, For a square matrix A of order 12345, if det(A)=1 and AA'=I (A' is the transpose of A) then det(A-I)=0 (I have to prove it if it is correct and provide a counterexample ...
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Best approximation of a vector $x=\begin{bmatrix}2&2&0&0\end{bmatrix}^\tau$ by the vectors in $M^\perp$, where $b\in M$ is given

In a unitary space $\Bbb R^4$, subspace $M=\operatorname{span}\left\{b=\begin{bmatrix}1\\1\\1\\1\end{bmatrix}\right\}\leqslant\Bbb R^4$ is given. Find the best approximation of the vector $x=\begin{...
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Does $A^TA=I$ imply $AA^T=I$?

Let $m,n\in\mathbb R^{m\times n}$ with $$A^TA=I_n\tag1.$$ I wonder whether this implies that $$AA^T=I_m\tag2$$ or if we can show it at least in the case $m=n$. EDIT: I was hoping for a proof which ...
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Opposite determinant in Autonne-Takagi factorization

Let us consider a complex symmetric matrix in $M_2(\mathbb C)$ \begin{equation} A = \begin{pmatrix} x_1+ix_2 & x_3 \\ x_3 & -x_1+ix_2 \end{pmatrix} \end{equation} where the $x_i\in \mathbb R,\;...
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How to show: Matrix $H_v = I - 2vv^{\top}$ with $||v|| =1$ to the reflection on hyperplane $v^{\bot}$ is symetrical and orthogonal and $det H_v = -1$?

I have several questions How can I show that a Matrix $H_v = I - 2vv^{\top}$ with $||v|| =1$ to the reflection on the hyperplane $v^{\bot}$ is symetrical and orthogonal and the determinant is $det ...
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Question about Orthonormal Bases

I solved this question but I am not quite sure if it is a valid solution. The question is: Let $u$ be a vector in an inner product space $V$ and let $\{v_1, v_2, v_3, \ldots, v_n\}$ be an ...
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A problem about operators on an Euclidean space

Let $A, B$ be self-adjoint positive operators, $C, D$ be orthogonal operators on an Euclidean space, and $AC = DB$. Prove that $C=D$. I know many properties of self-adjoint and orthogonal ...
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Calculate the matrix A of the linear transformation T: V→ V in the base S, the matrix C of change from the orthonormal base S to the orthonormal …

Calculate in the cases below (i) the matrix $A $ of the linear transformation $T: V \rightarrow V$ in the base $S \subset V$ and (ii) the matrix $C$ of change from the orthonormal base $S$ to the ...
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Operators on an Euclidean space

Let $A, B$ be self-adjoint positive operators, $C, D$ be orthogonal operators on an Euclidean space, and $AC = DB$. Prove that $A=B \Leftrightarrow AC$ is normal. I know many properties of ...
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For rotation matrices in SO(3), solutions of AB = BA given A (or B)

Suppose we have $\mathbf{AB} = \mathbf{BA}$, where $\mathbf{A},\mathbf{B} \in SO(3)$. What facts does this imply about $\mathbf{A}$ and $\mathbf{B}$? Clearly $\mathbf{A} = \mathbf{I}_3$ and $\mathbf{...
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How can I show that every vector in $U\oplus W$ can be uniquely decomposed into a sum of vectors in $U$ and $W$?

How can I show that every vector in $U\oplus W$ can be uniquely decomposed into a sum of vectors in $U$ and $W$? Let $V$be an inner product space over $\mathbb{R}$ (here, the inner product is denoted ...
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Singular value decomposition (SVD) for non-symmetric square real matrix contradicts spectral theorem?

From Bretscher's Linear Algebra with Applications: where $A$ is a real matrix in $ \mathbb{R}^{n \times m}$ and the singular values of $A$ are the square roots of the eigenvalues of the symmetric $A^...
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Invariant theory of the definite and indefinite orthogonal groups

I am vaguely aware of the following facts: Let $V$ be a finite-dimensional real vector space with a positive-definite inner product $g$. Let $g_{\otimes n}$ denote the natural extension of $g$ to $V^{...
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Obtaining a orthonormal matrix with 2 columns given

I have to create a orthonormal square matrix whose first row and last row, say $T_1$ and $T_N$ are given. Now what should be done to get the remaining columns of the matrix? The Gram-Schmidt ...
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A curious interrelationship between distinct embeddings of $SO(M+1)$ into $SO(2M+1)$

The following seems to be a property of $SO(2M+1)$ for an arbitrary integer $M$, although I have not yet been able to prove it. (I can prove it for, e.g., $M=1$, and have numerically checked it for ...
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Sphere is exhausted by orthogonal transformations of fixed vector

For each $ \ n \in \mathbb{N} \ $ let $ \, O(n+1) \, $ be the set of orthogonal linear operators $ \ \mathbb{R}^{n+1} \to \mathbb{R}^{n+1} \, $, $ \ e_{n+1} = (1,0,0,...,0) \in \mathbb{R}^{n+1} \ $ ...
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Rotation followed by reflection on 3D

How can I know the standard matrix of rotation and reflection, respectively from the standard matrix of rotation followed by reflection? $$ A=\begin{bmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ ...
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A symmetric matrix that is similar to a diagonal matrix [closed]

Perhaps someone here can help me with a homework exercise. Given a symmetric matrix $A$, find an orthogonal matrix $B$ such that $B^tAB=D$ is a diagonal matrix whose entries are arranged in ...
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A property of orthogonal matrix

suppose two orthogonal matrix $O_1, O_2 \in \mathbb{R}^{2 \times 2}$ in SO(2), consider $x = \text{Tr}(O_1O_2)$, we know that $$O_1 = \begin{pmatrix}\cos(\theta_1) &\sin(\theta_1) \\ -\sin(\...

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