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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Polar Decomposition of $O(p,q)$

I'm trying to show that in the polar decomposition $A= RL$ of $A\in O(p,q), \; p,q\geq 1$, $R\in O(p)\times O(q)$. Here I define $O(p,q) =\{A\in M(n,n,\mathbb{R})|A^TI_{p,q}A=I_{p,q}\} $, where $$I_{p,...
1 vote
1 answer
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Find orthogonal matrix $R$ that minimizes $\|R-Q\|_F$ for a complex Matrix Q

I have given a complex matrix $Q \in \mathbb{C}^{3,3}$ and I would like to find the "closest" rotation matrix (meaning $\mathcal{R}$ is orthogonal and $\det{(\mathcal{R})}=1$) $R \in \mathbb{...
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Can the notion of tangent space be recovered from an orthogonal projection?

Let $f:\mathbb{R}^D\to \mathbb{R}^p$ where $p=D-d$, let $J_f(x)$ be the $p\times D$ Jacobian of $f$ (we assume $f$ is at least $C^2$ smooth), and let $N(x)$ be the matrix obtained from ...
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Is there always a rotation matrix $U$, such that for a symmetric matrix $S$, $S=UKU^T$?

Assume that $U$ is a rotation matrix, i.e., $UU^T=I$ and $det\space U=1$. Is there always a $U$ such that for a symmetric matrix $S$, $S=UKU^T$, where $K$ is diagonal?
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Orthogonal diagonalization of dummy variables vectors covariance matrix

Let $n$ be a positive integer, and let $(p_i)_{i \in \{1,\cdots,n\}}$ be a finite sequence of real numbers, that are assumed to be nonnegative and sum to $1$. Let us denote by $\mathbf{p}$ the column ...
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Proof $I_{n} - \frac{2}{(\lvert v \rvert)^{2}} * vv^{t}$ is an orthogonal matrix

How do I proof that $I_{n} - \frac{2}{(\lvert v \rvert)^{2}} * vv^{t}$ is an orthogonal matrix, given that v $\in \mathbb{R}^{n}$ an non-trivial vector. I believe I could use the fact that $U * U^t = ...
1 vote
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Comparing the common matrix groups over $\mathbb{C}$

I am studying Lie groups and the book I am using starts off with matrix Lie groups. It states that the following matrix groups are of considerable importance (all over $\mathbb{C}$): $GL(n)$, the set ...
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Prove orthogonal projections for a matrix A and its transpose

I'm trying to prove that a projection is orthogonal using: $A ∈ \mathbb R^{m\times n} $ where $A^TA = I_n$, and I want to prove that $P=AA^T$ is an orthogonal projection. I understand that an instance ...
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For orthogonal matrices $A$ and $B$, prove $\det(A^{t}B - B^{t}A)=\det(A+B)\det(A-B)$

I can't prove this formula: $$ \det(A^{t}B - B^{t}A)=\det(A+B)\det(A-B) $$ I tried using fact that $A^{t}A = I$ (similarly for $B$): $$ A^{t}B-B^{t}A=A^{t}B-B^{t}A + A^{t}A - B^{t}B=A^{t}(A+B)+B^{t}(A ...
1 vote
1 answer
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ij-element of (A^T)A matrix

I know that given an orthogonal matrix $A\in\mathbb{R}^{n\times n}$, $A^\top A=AA^\top=I_n$. I saw that the $ij$-element of $A^\top A$ can be expressed as $$(A^\top A)_{ij} = (A^\top a_j)_i=(\text{row ...
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Orthogonal Diagonalization; how does A = D?

My textbook says to confirm that: $$P^TAP = \begin{bmatrix}-1/\sqrt{2}&1/\sqrt{2}&0\\-1/\sqrt{6}&-1/\sqrt{6}&2/\sqrt{6}\\1/\sqrt{3}&1/\sqrt{3}&1/\sqrt{3}\end{bmatrix} \begin{...
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What is the spectral family of $ \left[ \begin{array}{cc} 0&1&0\\ 1&0&0\\ 0&0&1 \end{array} \right] $?

Spectral Family:A real spectral family (or real decomposition of unity) is a one parameter family $\mathcal E=(E_{\lambda})_{\lambda\in \mathbb R}$ of projections $E_{\lambda}$ defined on a hilbert ...
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Why can the Orthogonal group be split up in this way?

In my groups course at university, we’ve spent a while on the orthogonal group, leading up the the conclusion that $$ \mathrm{O}_n = \mathrm{SO}_n \mathbin{\dot{\cup}} \begin{pmatrix} -1 ...
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I'm getting that orthogonal matrices don't (necessarily) have orthogonal rows/columns

As the title says, here's my "proof": Let U be some orthogonal matrix: Uᵀ = U⁻¹ ∴ U Uᵀ = Uᵀ U = I Considering the ijth element: (U Uᵀ)ᵢⱼ = (Uᵀ U)ᵢⱼ = δᵢⱼ ∑UᵢₖUᵀₖⱼ = ∑UᵀᵢₖUₖⱼ = δᵢⱼ ∑UᵢₖUⱼₖ = ∑...
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4 answers
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Prove that the eigenvectors of a square real matrix A are orthogonal if and only if ${A^T}A=A{A^T}$ [duplicate]

Note that this is more general than the usual orthogonal matrix has orthogonal eigenvectors or symmetric matrix has orthogonal eigenvectors in that $A$ need not be orthogonal or symmetric, just square....
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Generalizing the hypersurface formula for orthogonal projection matrices

Let $M=f^{-1}(\{0\})$ be a $d$ dimensional manifold situated in $\mathbb{R}^D$, where $D>d$. We assume $f:\mathbb{R}^D\to \mathbb{R}^p$ where $p,D,d$ are related by $p=D-d$. For those interested in ...
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Orthogonal Matrix Conditions

I am a bit confused about a matrix being orthogonal. A square matrix $A$ is said to be orthogonal if $AA^T = I$, and also if $A^T = A^{-1}$. Now I was looking through the internet and found a third ...
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What sets of orthogonal matrices $A$ and $B$ satisfy $AABBAB=BBAABA$? [closed]

Given two continuous curves of orthogonal matrices $t\to A_t$ and $ t\to B_t$, such that for any value of $t$ we have: $AABBAB=BBAABA$ $A$ and $B$ do not commute. What solutions are there for $A_t$ ...
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For orthogonal matrices $A$ and $B$, does $ABAB=BABA$ imply $A$ and $B$ commute?

Given two distinct orthogonal matrices $A$ and $B$, given some individual sequence of applications of these matrices such that each matrix appears an equal number of times (e.g. $AABBAB$, with $3$ ...
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Existence of orthonormal matrix

I am looking for an orthonormal matrix $A\in\mathbb{C}^{n\times n}$, such that for any two elements $a_{i\ j},a_{p\ q}$ ( not necessarily the distinct elements ) of $A$, we have that: \begin{equation} ...
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What should be the value of $p,q,r,s,$ and $t$ so that the following matrix is orthogonal?

What should be the value of $p,q,r,s,$ and $t$ so that the following matrix is orthogonal? $$\left[ {\begin{array}{cc} p & q & r \\ \frac{1}{\sqrt{3}} & q & s \\ \frac{1}{\sqrt{3}} &...
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Action that involves a rotation on the real hyperplane

For an element $v \in V$ with $d(v,v) = 1$ (i.e. $v$ is a unitary vector), define $r_v \in \mathrm O(V)$ by $$ r_v(w) := -\rho_v(w) := 2d(v,w)v - w. $$ Here $\rho_v$ is the reflection in the real ...
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Extension of Wahba’s problem with infinite weights

Wahba's problem seeks to find a $3 \times 3$ orthonormal rotation matrix that minimizes $$ J(\mathbf{R}) = \frac{1}{2} \sum_{k=1}^{N} a_k\| \mathbf{w}_k - \mathbf{R} \mathbf{v}_k \|^2 $$ I want to ...
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Prove existance of orthonormal basis of $\mathbb{R}^3$ consisting of eigenvectors of generalized eigenvalue equation

Context I am studying normal modes oscillations and normal modes [1,2]. In an earlier post [3], I asked for a proof that the generalized eigenvalue equation in normal-mode analysis has positive ...
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Row (or column) sums of orthogonal matrix (excluding scalar multiples of identity matrix) always different

Are row (or column) sums of orthogonal matrix (excluding scalar multiples of identity matrix) always different? Suppose $Q$ is an $p\times p$ orthogonal matrix that is not a scalar multiple of ...
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Is there are a $3 \times 3$ real, orthogonal matrix $Q$ that has exactly three zero entries? [closed]

If I have choose first column with two zeros, then other two columns have two or none. Or else there are are three zeros in each column. I don't know how to conclude after this
2 votes
1 answer
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If an orthogonal matrix represents a reflection, show that it is symmetric. [closed]

The question is as it says. I am a first-year uni undergraduate student for context. I don't really know how to approach this question. Any hints/suggestions would be greatly appreciated!
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Is the external direct product of O(2) with itself isomorphic to O(2)? (Orthogonal group of 2 x 2 M-matrices)

Is the external direct product of $O(2)$ with itself isomorphic to $O(2)$ i.e. $O(2) \oplus O(2) = O(2)$ I'm guessing they're not, as to prove that is was, I'd have to define an explicit isomorphism ...
2 votes
1 answer
348 views

Prove that [O(2), O(2)] = SO(2). The commutator subgroup of the ortogonal group is equal to the special orthogonal group.

Prove that $[O(2),O(2)] = SO(2)$. In words, prove that the commutator subgroup of the orthogonal group of 2x2 matrices is equal to the the Special Orthogonal Group of 2x2 matrices. I know that the ...
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Inconsistency with multiplication of different size of orthogonal matrices

I am a little confused about the following: Let A be an $\mathbb{R}^{n \times f}$ matrix, B be an $\mathbb{R}^{m \times f}$ matrix, and C be an $\mathbb{R}^{f \times d}$ orthogonal matrix. If $f > ...
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Show that $\mathrm{O}(q)$ is the union of a set of vector symmetries and the set of applications $\gamma_a$

We fix $q$ the quadratic form on $\mathbb{R}^2$ given by $q\left(x_1, x_2\right)=x_1 x_2$. Let $$\mathrm{O}(q)=\left\{f \in \mathrm{GL}_2(\mathbb{R}) \mid q(x)=q(f(x)) \text { pour tout } x \in \...
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Finding an orthogonal basis of a union of two subspaces quickly

Let's say we have two row-orthonormal complex matrices $P \in \mathbb{C}^{n \times k}$ and $Q \in \mathbb{C}^{m \times k}$, $n + m < k$. Since those martices are row-orthonormal, rows of $P$ form ...
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Are all matrices $A$ with $A^T=A^{-1}$ permutation matrices?

I know that all permutation matrices $A$ satisfy $A^T=A^{-1}$, but is the converse, all matrices $A$ with $A^T=A^{-1}$ are permutation matrices, also true? I believe it is true, but I haven't been ...
4 votes
1 answer
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rotationally invariant matrix function

Consider a function $f:\mathbb{R}^{N \times M} \to \mathbb{R}^{N \times M}$, that takes a matrix $\mathbf{A} \in \mathbb{R}^{N \times M}$ as input and the output is a matrix of the same size. Suppose ...
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Preservation of angles

In this post, to prove that left multiplying a matrix $V$ by an orthogonal matrix $A$ preserves angles among columns of $V$, the author used the following equality: $(Av_i)^T(Av_j)=v_i^TA^TAv_j = v_i^...
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Exercise 4, Section 5.4 of Hoffman’s Linear Algebra

An $n\times n$ matrix $A$ over a field $F$ is called orthogonal if $AA^t = I$. If $A$ is orthogonal, show that $\text{det}(A)=\pm 1$. My attempt: By theorem 3 section 5.3, $\text{det}(AA^t)=\text{det}...
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Relationship between rows and columns of an orthogonal matrix in a spectral decomposition

I am trying to derive the relation $\mathbf A=\mathbf V^T\boldsymbol\Lambda\mathbf V=\sum_{i=1}^n\lambda_i\mathbf v_i\mathbf v_i^T$, where $\mathbf A$ is symmetric, $\mathbf V$ orthogonal (where $\...
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1 vote
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Quadratic form and orthogonality

If the equation $\textbf{x}^T \textbf{x} = \textbf{x}^T M \textbf{x}$ is true for all $\textbf{x}$ and a matrix M in a euclidian vector space, does that imply that M is an orthogonal matrix? And if so,...
1 vote
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Orthogonal projection of a matrix onto the line

I have the following exercise: Let $V = W = \mathbb{R}^3$ and let $f$ be the mapping given by the orthogonal projection onto the line passing through the points (0, 0) and (3, 2). Let E = {$e_1$, $e_2$...
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Non-square orthogonal matrix $B$ with $B^TB=I$

Let $B\in\mathbb{R^{p\times q}},p\geq q$ such that $B^TB=I_q$, is it true that all the diagonal elements of $BB^T$ are all no greater than 1? How can I prove that?
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Show that a matrix has a Cholesky factorization providing that it can be written as a product of a matrix and its transpose [duplicate]

$A$ is an invertible real square matrix ($A \in \mathbb{M_{n}(\mathbb{R})}$ and $det(A) \neq 0$). Let's consider another matrix $B \in \mathbb{M_{n}(\mathbb{R})}$ such that: $$B = {}^\intercal A \cdot ...
1 vote
1 answer
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A skew symmetric and orthogonal matrix has eigen values (3/5) + (4i/5). How can this be possible? It must have 0 or purely imaginary values. Problem 1

Problem 1. It is Orthogonal and skew symmetric but eigen values aren't purely imaginary or zero Are the following matrices symmetric, skew-symmetric and/or orthogonal? $$\frac15\begin{bmatrix}3&-4\...
4 votes
1 answer
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Maximizing $\mbox{tr} \left( {\bf X}^{-1} {\bf A} {\bf X} {\bf B} + 2 {\bf X} {\bf C} \right)$ subject to ${\bf X} {\bf X}^\top = \gamma {\bf I}$

Can the following optimization statement be converted into a simpler form? $$\begin{array}{ll} \underset{{\bf X}}{\text{maximize}} & \mbox{tr} \left( {\bf X}^{-1} {\bf A} {\bf X} {\bf B} + 2 {\bf ...
2 votes
3 answers
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3D rotation matrix question: Extension of Wahba's Problem

This is an extended version of Wahba's Problem. Let $R_x(\alpha)$ denote a 3D rotation matrix around the axis $x$ by an amount $\alpha$. For given unit vectors $\{u_k\}_{k=1}^3$ and $\{v_k\}_{k=1}^3$ ...
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Understanding why the inverse of an ortogonal matrix is its transpose

I have seen this question asked many times here for example: Why is inverse of orthogonal matrix is its transpose? I understand how the answers on that question proves it. But I want to understand it ...
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Necessary and sufficient conditions for an existence of an orthogonal matrix$P~$ s.t. $~P^{-1}AP~$is diagonal, using$~a~$which is one of entries of$A$

This problem is quoted from the $3$rd year transfer exam of math major in the university. $$\begin{align} a:=\text{real number}\\ A:= \begin{pmatrix} 0&a&2\\ 1&0&2\\ 2&2&3 \end{...
5 votes
2 answers
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The convex hull of rotations does not contain reflections

$\newcommand{\SO}{\operatorname{SO}_n}$ $\newcommand{\Om}{\operatorname{O}_n^{-}}$ I saw here the following claim: Let $\SO$ be the special orthogonal group, and let $\Om$ be the orthogonal matrices ...
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Degree of the switch map

I am trying to show the following lemma: Lemma: Let $n,m \geq 1$ and consider the switch map $s: S^m \wedge S^n \to S^n \wedge S^m$, given by $[(x,y)] \mapsto [(y,x)]$. Then we have $\text{deg}(s)=(-...
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1 vote
1 answer
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Calculate rotation matrix to rotate a matrix A (3d points x,y,z) to be orthogonal by 3D normal vector N [closed]

I have a matrix A, which is the coordinates of a circle in 3D space. I want to rotate the circle in a way that its normal vector (orthogonal to the circle) be aligned with vector N:(x_n,y_n,z_n). I'd ...
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Non-similitudes which preserve relative similarity

What maps $h:\mathbb{R}^N\to\mathbb{R}^N$ exist such that for $A,B\subset\mathbb{R}^N$ we have $A\sim B \iff hA\sim hB$. There is a trivial solution when $h$ is a similitude: $h(x)=rOx+t$ where $r\in \...
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