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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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How many degrees of freedom do orthogonal skew-symmetric matrices have?

$n$ by $n$ real orthogonal matrices have $n (n-1)/2$ degrees of freedom. So do the skew-symmetric matrices. But what about matrices that are both skew-symmetric and orthogonal? Is the number of such ...
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$3\times 3$ orthogonal matrix, which doesn't consist of zeros and ones [closed]

I'm stuck with my homework in a subject called Matrices in Statistics. Can you guys help with the following task? I would be very thankful! The task is as follows: Find a $3\times 3$ orthogonal ...
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2answers
687 views

Show non-singularity of orthogonal matrix

I'm given a question that says: A matrix $Q$ of size $n \times n$ is called orthogonal if its columns are orthogonal to each other and all columns have length $1$. a) Show that the matrix is non-...
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1answer
45 views

Every hyperplane contains an orthogonal matrix

Let $E$ be an euclidean space (over $\mathbb{R}$), I have to prove that every hyperplane of the linear maps over $E$ contains an orthogonal map (or equivalently, matrix). What I've tried doing is ...
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335 views

Which statement is false ?(Linear algebra problem)

Let $P=\dfrac{xx^{T}}{x^{T}x}$ be an a square matrix of order n where $x$ is a non zero column vector. Then which one of the following statement is False. $(A)$ P is idempotent $(B)$ P is ...
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42 views

Finding An Orthogonal Transformation Matrix

I have two symmetric matrix 5x5 ,A and B that A=$\begin{bmatrix}a11&a12&a13&a14&a15\\a12&a22&a23&a24&a25\\a13&a23&a33&a34&a35\\a14&a24&a34&...
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46 views

Question involving orthogonal matrix and congruent matrices $P^{t}AP=I$

Suppose $A$ is a real, symmetric, positive matrix, s.t. $P^{t}AP=I$ for some congruention matrix $P$. Prove that $\forall Q\in M_{n}(\mathbb R), Q^{t}AQ=I\iff Q=PU$, where $U$ is orthogonal. I can't ...
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Rotating a matrix to become symmetric

Given an $n \times n$ matrix Q having eigenvalues in $(0,1],$ is it possible to find an $n \times n$ orthogonal matrix $U$ such that $$(QU)^T = QU$$ holds and the eigenvalues of $QU$ also fall in $(0,...
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2answers
137 views

A property of orthogonal matrices

Let $R$ be a $3\times 3$ orthogonal matrix. Let $v$ be the unit vector such that $Rv=v$ (upto sign change). Consider any unit vector $u$ such that $u^{T}v=0$ where $T$ stands for transpose. Show that ...
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30 views

Functions on $\mathbb{R}^n$ commuting with orthogonal transformations [duplicate]

I am interested in finding the functions $f:\mathbb{R}^n \to \mathbb{R}^n$ for which $f \circ U = U \circ f$ for all orthogonal transformations $U:\mathbb{R}^n \to \mathbb{R}^n$. Note that $f$ need ...
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Optimization over images of column-orthogonal matrices through rotations and reflections

The sets of orthogonal matrices with determinant +1 ("rotations") and -1 ("reflections"), respectively, make up distinct sets, so that if we label the sets $R$ and $M$, minimizing $$ \min_{O\in R}\|O^...
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1answer
50 views

Minimum of the 2-norm

Let $A = QR$, where $Q$ is an orthogonal ($m\times m$)−matrix and $R$ is an upper ($m\times n$)-triangular matrix of rang $n$ ($m>n$). I want to show that $$\min_{x\in \mathbb{R}^n}\|Ax-y\|_2=\|(...
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21 views

Isotropic tensor field depending a vector

I am wondering how to prove the following statement (which is widely used, for example in turbulence theory) mathematically rigorously: Assume we are talking about $V=\mathbb{R}^3$. Given a tensor ...
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88 views

An orthogonal matrix which sends a vector to other vector with same length.

I know that For every to vectors $u,v\in \mathbb R^n$ where $|u|=|v|$ there exists an orthogonal matrix $A$ such that, $Au=v$. I have a problem so construct this matrix. is there any method to ...
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1answer
113 views

Deriving the Optimal Solution of the Orthogonal Procrustes Problem

I am trying to work through the Orthogonal Procrustes Problem but I do not understand a particular step. I would appreciate any help in understanding the steps the author goes from the first line to ...
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1answer
38 views

Find a Orthogonal Transformation that sends a plane into another

Consider $\mathbb{R}^n$. Define $U=\{x\in\mathbb{R}^n|x=\lambda_1u_1+\lambda_2u_2, u_i\in\mathbb{R}^n, \lambda_i\in\mathbb{R}\}$ and $V=\{x\in\mathbb{R}^n|x=\lambda_1v_1+\lambda_2v_2, v_i\in\mathbb{...
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62 views

How to orthogonally project to boundary of the ball $\mathcal{B}_r(0)\triangleq\{Y\in\mathbb{R}^{n\times n}:\|Y\|_2<r\} $?

Let $r>0$, and let $A\notin\mathcal{B}_r(0)\triangleq\{Y\in\mathbb{R}^{n\times n}:\|Y\|_2\leq r\}$, where $\|\cdot\|_2$ is the induced 2-norm. Let $\bar A$ be the orthogoanl projection of $A$ on ...
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757 views

Show that this matrix is singular [closed]

If $\det P=-1$ and $P$ is an orthogonal matrix. Show that $P+I_n$ is singular matrix. Please help it with only matrix algebra.
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Orthogonal matrix pairs connected by rescaling of rows and columns

For which choices of column-orthogonal matrix $U\in\mathbb{R}^{m\times n}$ and real diagonal matrices $\Lambda_0$ and $\Lambda_1$ (different from the identity) does it hold that $$ O=\Lambda_0U\...
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3answers
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Find a rotation matrix that sends $v$ to $u$

I know we have closed formulas for $\mathbb{R}^3$. I am looking for arbitrary $N$. Given $\mathbb{R}^N$, find an orthogonal matrix $U$ that sends a unit-vector $u$ to unit-vector $v$. Is there a ...
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1answer
52 views

Every linear orthogonal transformation can be represented as a block matrix

How would you show that every linear orthogonal transformation can be represented as a block matrix, with the blocks either being 2x2 rotation matrices or $\pm1$? I have managed to show it for the 2 ...
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2answers
610 views

Prove that a self adjoint and idempotent matrix is a orthogonal projection matrix.

$\newcommand{\R}{\operatorname{Ran}} \newcommand{\K}{\operatorname{Ker}}\newcommand{\b}{\mathbf}$ Prove that a self adjoint and idempotent matrix $P$ is an orthogonal projection matrix. I was ...
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1answer
170 views

Gram-Schmidt orthogonalization: Dealing with Complex numbers

A complex valued matrix "A" has n columns a_1 through a_n. Elements of these columns are complex numbers. The orthogonal complex valued matrix U of A has n columns as well u_1 through u_n. u_1 is ...
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Show that the special orthogonal group $SO_n(\mathbb{R})$ is arcwise connected and compact

I'm doing the following exercise: Using that we know that if $M\in SO_n(\mathbb{R})$ there exists a $P\in GL_n(\mathbb{R})$ such that $M=PM'P^{-1}$, where $M'$ has zeros on everywhere except on its ...
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1answer
201 views

Maximize function on orthogonal matrices

Consider the function $f$ from the set of $n \times n$ real matrices taking $A=(a_{ij})$ to $f(A):= \prod_{(i,j) \neq (k,l)}(a_{ij}-a_{kl}) $. Edit: Note that $f(A) \ge 0$ for all $A$, since grouping ...
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1answer
93 views

Orthonormal matrices and spectral norm

Let $\mathbf{X}\in \mathbb{R}^{n\times d}$ be a matrix with orthonormal columns ($d\le n$). By sub-multiplicativity, we have that $\|\mathbf{A} \mathbf{X}\|_2 \le \|\mathbf{A}\|_2$ and $\|\mathbf{X} \...
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1answer
59 views

Finding a special orthogonal matrix to generate pentadiagonal matrix

Suppose a Jacobian matrix $A_{n \times n}$ is given. I need to find an orthogonal matrix $Q_{n \times n}$ such that $Q^T A Q = B$ and $B$ is a pentadiagonal matrix. I need to know if this problem can ...
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1answer
55 views

Linear algebra proof on orthogonal diagonalization

Show that if an $n × n$ matrix $A$ is positive definite, then there exists a positive definite matrix $B$ such that $A = B^tB$. The way I set out to show this was (note that P is an orthonormal ...
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1answer
34 views

Values of $n$ so that exist a matrix $A\neq 0$ so that $Ax$ is orthogonal a $x$ for all $x\in \mathbb{R}^n$.

Find the values of $n$ so that exist a matrix $A\neq 0$ with reals entries so that $Ax$ is orthogonal a $x$ for all $x\in \mathbb{R}^n$. I try solved this exercise using the theory of orthogonal ...
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0answers
75 views

Prove that a real $2 \times 2$ matrix is orthogonal if and only if it is of one of the following forms: Proof Review And Possible Author Error?

Prove that a real $2 \times 2$ matrix is orthogonal if and only if it is of one of the forms $\begin{bmatrix} a & b \\ -b & a \end{bmatrix}$, $\begin{bmatrix} a & b \\ ...
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1answer
838 views

Can we prove that the eigenvalues of an $n\times n$ orthogonal matrix are $\pm 1$ by only using the definition of orthogonal matrix? [duplicate]

Can we prove that the eigenvalues of an $n\times n$ orthogonal matrix are $\pm 1$ from the definition of orthogonal matrix alone? An $n\times n$ matrix $A$ is orthogonal iff $AA^T=A^TA=I$. Is it ...
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1answer
99 views

prove that the symmetric projection is always orthogonal. [closed]

prove that the symmetric projection is always orthogonal. answer : we have: null(A)=ran(At) then A=At so null(A)=ran(A)
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Show that the matrix $P=I-2hh^T$ is orthogonal and find its first column.

Let $x=(x_1,...,x_n)^T$ a column vector in $\mathbb{R}^n$ so that $x_1\neq -1.$ Let $h$ a unitary vector in the direction of $x-e_1$ where $e_1$ is the vector in $\mathbb{R}^n$, $e_1=(1,0,...,0)$. ...
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1answer
235 views

Proving subgroup of $SO(2)$ with $n$ elements is unique?

I was thinking of taking two subgroups with $n$ elements and showing that they both are the same subgroup. I know that if $M$ ($2\times2$ matrix) is any element of $SO(2)$, $M^{-1} = M^t$. And $\...
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1answer
124 views

Condition number is less than n

Show that for an $n$ x $n$ orthogonal matrix $A$ that $\operatorname{Cond}(A) \leq n$. I need to use: $$\|x\|_1 \leq \sqrt n$$ I know that $\operatorname{Cond}(A)=1$ for $A$ orthogonal matrix. ...
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0answers
107 views

Show that for $n \times n$ orthogonal matrix $A$ that $\operatorname{Cond}(A)\leq n$ [closed]

Show that for $n \times n$ orthogonal matrix $A$ that $\operatorname{Cond}(A)\leq n$ How do I start with this question? Do I relate this to rank-nullity theorem?
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1answer
35 views

Existence of a particular matrix $A$ $\in \mathrm{SO}(\mathbb{R}^n)$

I need to prove that for all $n\in \mathbb{N}$ exists a matrix $A$ $\in \mathrm{SO}(\mathbb{R}^n)$ (orthogonal matrixes with determinant equal to 1) such that the inputs of the first row are equal to $...
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1answer
73 views

Subspaces invariant under orthogonal similarity transformations

Let $\mathcal{S}_n$ denote the vector space of all real symmetric $n \times n$ matrices. Is there a characterization of the subspaces $V$ of $\mathcal{S}_n$ that are invariant under orthogonal ...
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1answer
249 views

Are there any shortcuts to tell if a square matrix is orthogonal?

So, if one is asked if a given matrix $A$ is symmetric, one could compute $A^T$ and check if $A^T=A$, however you can also simply check the symmetric entries accross the diagonal and see if they are ...
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1answer
68 views

“Completing” an LQ decomposition

Let $A$ be a real $m\times n$ matrix with $n>m$. Let $Q_1$ be a $n\times m$ with orthonormal columns such that $$ AQ_1 = L $$ where $L$ is of dimension $m\times m$ and lower triangular. Question:...
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1answer
47 views

Doubts on obtaining orthonormal basis

In finding the matrix $P$ that orthogonally diagonalizes $A$ and to determine $P^TAP$, where $$A = \begin{pmatrix} 1 & -1 & 1 & -1\\ -1 & 1 & -1 & 1\\ 1 & -...
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198 views

How can I prove that RGB color space is or is not an Orthogonal Transformation?

I was searching a lot of information about Color Space and how can I prove that the RGB color space is or not Orthogonal Transform, I suppose that it's Orthogonal Transform because all the image of ...
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92 views

Geometry of $Sp(2N,R)/U(N)$

Given $\mathbb{R}^{2N}$ equipped with a symplectic form $\Omega^{ab}$ and a compatible symmetric, positive definite, bilinear form $G^{ab}$, we can look at the symplectic group $\mathrm{Sp}(2N,\mathbb{...
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311 views

Show when the inequality for matrix-vector multiplication for the 2 norm is an equality?

I am asked to show for which vectors the inequality $||Ax|| \le ||x||||A|| $ is an equality. My intuition tells me that this happens when $x$ is in the direction of the right singular vector ...
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1answer
389 views

Why eigenvalues of an orthogonal matrix made with QR decomposition include -1?

I want to make a real orthogonal matrix whose eigenvalues don't include -1. However, eigenvalues of a matrix $Q \in \mathbb{R}^{n\times n}$ ($n$ is even number) made with QR decomposition of a random ...
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2answers
38 views

Let $A(\theta)$ be a given function , where $\theta \in (0, 2\pi)$. Mark the correct statement below

Let $ A(θ) = \left[ {\begin{array}{cc} \cosθ & \sinθ \\ -\sinθ & \cosθ \\ \end{array} } \right] $ where $θ ∈ (0, 2π)$. Mark the correct statement below A. $A(θ)$ has ...
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1answer
63 views

4 dimensional orthogonal group

Let's say I have a 4 by 4 matrix that is in the orthogonal group. The first three columns A, B, and C are known. Now, I can do a system of equations (4 equations) to solve for D, the fourth column. ...
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1answer
53 views

Orthogonal Group

(Note: $O(n) = O(n, \Bbb{R})$) I've notice that $O(1)$ is equivalent to $S^0$. And I've read that $O(2)$ is equivalent to two copies of the circle group $S^1$. I was wondering if someone can ...
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1answer
176 views

Nearest Semi Orthonormal Matrix Using the Entry Wise $ {\ell}_{1} $ Norm

Given an $m \times n$ matrix $M$ ($m \geq n$), the nearest semi-orthonormal matrix problem in $m \times n$ matrix $R$ is $$\begin{array}{ll} \text{minimize} & \| M - R \|_F\\ \text{subject to} &...
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1answer
685 views

If A and B are unitarily equivalent, then they have the same singular values using uniqueness?

I'm trying to prove the following statement: if A and B are unitarily equivalent, then they have the same singular values so my proof goes like this: unitarily equivalent means $A=QBQ^*$ so if $A=...