Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
1answer
37 views

Are the following two sets of rank one, trace one, PSD matrices isomorphic?

Consider the following two sets of matrices: The first set is produced by $vv^T$ with $v = [v_1 \ \ v_2]^T$ $$\left\{\begin{bmatrix} v_1^2 & v_1v_2 \\ v_1v_2 & v_2^2 \end{bmatrix}: v_1^2+...
2
votes
1answer
36 views

Rigorous way to show a continuous function of matrices is orthogonal

Suppose $Q(t)$ is a continuous set of real matrices with $t \in \mathbb R$ a parameter. Suppose $Q(0)$ is orthogonal and suppose $$Q'(t) = S(t) Q(t)$$ where $S(t)$ is a continuous set of real skew-...
1
vote
1answer
932 views

Convex hull of rotation matrices is closed and contains the origin

I am reading the paper Semidefinite descriptions of the convex hull of rotation matrices by James Saunderson, Pablo A. Parrilo and Alan S. Willsky. On page 2, it says: I "guess" the set of ...
2
votes
2answers
102 views

Multiplicativity of “isometric projections” and commutation of matrices

Every $A \in \text{GL}_n^+(\mathbb{R})$ (an invertible matrix with positive determinant) has a unique Polar decomposition: $A=OP$ where $O \in \text{SO}_n$, $P$ symmetric positive definite. The ...
4
votes
1answer
189 views

What can be said about a convex combination of orthogonal matrices?

Let $A$ and $B$ be two orthogonal matrices (of order $n\geqslant 2$) such that $$\det A=1 \qquad\qquad \det B=-1$$ Can we say that: there is $\lambda \in [0,1]$ such that $\lambda A +...
4
votes
1answer
209 views

What is the equation of the orthogonal group (as a variety/manifold)?

I have been studying some elementary Lie theory recently, so I have been thinking about matrix groups as manifolds. Most simple examples of manifolds that we learn in high school or college even are ...
0
votes
0answers
80 views

Isomorphism between $O(2n,\mathbb{R})$ and $O(n,n,\mathbb{R})$ and same question for their Lie algbera

Is there any isomorphism of Lie groups between $O(2n,\mathbb{R})$ and $G:=\{ X \in M_{2n\times 2n}(\mathbb{R}) \mid X^tSX = S\} $ where $S$=$\begin{pmatrix} & {I_n}\\ I_n & \end{pmatrix}$....
1
vote
1answer
213 views

Least squares solutions and orthogonal projection?

I found the least squares solution for the following inconsistent system of equations: $ x_1 - x_2 = 0$ $ x_1 + x_2 = 5 $ $-x_1 + x_2 = 2$ , which turned out to be $ \begin{bmatrix} 2\\ 3\\ \end{...
1
vote
1answer
69 views

Calculating $\|A\|_2$ in terms of eigenvalues of $A^\ast A$

Let $A$ be a real matrix. I'm supposed to calculate $\|A\|_2$ in terms of the eigenvalues of $A^t A$. I thought to just diagonalize $A^t A$ as $UD^2U^t$ but then I have $\|Ax\|_2 =x^tUD^2U^tx$ instead ...
0
votes
0answers
54 views

Is a matrix that is orthogonally diagonalizable a projection matrix?

If a have a matrix say $A$ that is orthogonally diagonalizable (i.e. it can be written as $\lambda_1u_1u_1^T+ \lambda_2 u_2u_2^T+\dotsc \lambda_nu_nu_n^T$ , where the $u_i$ are the eigenvectors of the ...
0
votes
0answers
165 views

Is there a characterization of linear isomorphisms of the space of skew symmetric matrices?

Let $M_n^s$ denote the $\scriptstyle\binom{n}{2}$ dimensional space of $n \times n$ skew-symmetric matrices. Is there a characterization of linear isomorphisms that take $M_n^s$ into itself. If $n=2$ ...
1
vote
0answers
32 views

Polar of orthogonal set invariant under group action

I just ask the following question: Set invariant under group action Furthermore, How to prove the green part Original paper: http://arxiv.org/pdf/1403.4914v1.pdf (p.1324) Let $$O(n)=\{...
2
votes
0answers
248 views

Set invariant under group action

I am reading a paper with the following description: $O(n): \{Y\in \mathbf{R}^{n\times n}\mid Y^TY=I\}$ We say a set $V$ is $T$-invariant if $TV\subseteq V$, where $T$ is a linear transform. So ...
2
votes
0answers
100 views

Norms (eigenvalues) of sums of orthogonal matrices

Let $T_1, \ldots, T_n$ be a set of real-valued symmetric matrices satisfying $Tr(T_j T_k) = 0$ for all $j\neq k$. Consider the norm $\|T\|_{\infty} = \max_{\|M\|_1 \leq 1} \operatorname{Tr}\left[M^T ...
1
vote
0answers
109 views

Proof of the rotation matrix is an extreme point of $\text{conv } SO(n)$

Define the set of rotation matrices: \begin{equation} \begin{aligned} SO(n) := \{X\in \textbf{R}^{n\times n}: X^TX=I, \text{det}(X)=1\} \end{aligned} \end{equation} I want to prove that if $X\in SO(...
2
votes
1answer
756 views

Improper rotation matrix in $2D$

The following is the related problem: Improper Rotations in Even Dimensions I want the simpler explanation. An improper rotation is rotation, followed by reflection in the plane perpendicular ...
0
votes
2answers
57 views

Are the groups $G:=\{A \in M(n,\mathbb R) : A=A^t\}$ i.e. the group ( under addition ) of symmetric matrices and $O(n,\mathbb R)$ isomorphic?

Let $G:=\{A \in M(n,\mathbb R) : A=A^t\}$ i.e. the group ( under addition ) of symmetric matrices ; Are $G$ and $O(n,\mathbb R)$ isomorphic ?
0
votes
0answers
190 views

Finding an orthogonal matrix for a 3x2

I know how to find an orthogonal matrix for a $2\times2$ or $3\times3$ matrix. However I have been stuck on how to do this for a $3\times2$ matrix. The question is how to find a non-zero $3\times2$ ...
3
votes
1answer
2k views

What exactly does a rotation preserve?

I understand a rotation should preserve length and angle and hence the dot product. Since anything that preserves the dot product is a linear transformation, then a rotation can be represented by a ...
4
votes
2answers
2k views

For an orthogonal matrix $Q$, why does $QQ^T = I$?

In my linear algebra text (Strang), an orthogonal matrix is defined to be a square matrix whose columns are orthonormal. In other words, an orthogonal matrix is a matrix $Q = [q_1 \cdots q_n]$ where ...
2
votes
1answer
76 views

If a matrix is such that its inverse is equal to its transpose, does it belong to the special orthogonal group

I'm trying to solve 3 problems on special orthogonal groups, and I need proof verification of the first 2 and help with the proof of the 3rd. Consider $SO(n)$ the set of all $n \times n$ matrices ...
4
votes
2answers
808 views

Show any orthogonal matrix is similar to an almost diagonal matrix, with either $\pm 1$ or a 2D rotation on the diagonal

Let $A \in O(n).$ Show that $A$ is similar to a matrix which consists of $2 \times 2$ blocks down the diagonal of the form $$ \begin{pmatrix} \cos{\theta} & \sin{\theta}\\-\sin{\theta} & \cos{...
0
votes
1answer
480 views

Orthogonal matrix of symmetric matrix (eigenvectors)

I have this exercise: First of all, I can't find such $v_3'$ such that $v_2 · v_3 = 0$. Second, how is the orthogonal matrix $P$ when the normalized vectors $v_1, v_2$ and $v_3'$ have different ...
2
votes
3answers
68 views

Prove that if $A^t = A^{-1}$ then different rows of $A$ are orthogonal to each other

Prove that if $A^t = A^{-1}$ then different rows of $A$ are orthogonal to each other I saw this statement in my linear algebra book and I don't really know how to prove it, I hope someone can help me ...
1
vote
2answers
170 views

Show that $x$ and $Q x$ are equidistant

$$Q= \begin{bmatrix} \cos x & -\sin x\\ \sin x & \cos x\end{bmatrix}$$ Given x belongs to $\mathbb{R^2}$, show $Qx$ and $x$ are equidistant. I've tried dot producting $Qx$ and seeing whether ...
2
votes
1answer
96 views

Orthogonal matrix $Q$ such that $\forall x\leq 0$, $Qx\geq 0$

What are the orthogonal matrices $Q$ such that for all vectors $x\leq 0$, $Qx\geq 0$? The inequality is to be understood component-wise. In dimension 1, the only possibility is $Q=[-1]$, which is a ...
2
votes
1answer
1k views

Show that $O(n)$, the set of orthogonal $n \times n$ matrices, is not connected

I want to show that $O(n)$, the set of orthogonal $n \times n$ matrices is not connected. I know that a connected space $X$ does not split into disjoint non-empty open subsets, so to prove $O(n)$ is ...
2
votes
1answer
635 views

If $\lambda$ is an eigenvalue of an orthogonal matrix $C$, prove $\frac{1}{\lambda}$ is an eigenvalue of $C^T$

$C$ is an orthogonal matrix. If $\lambda$ is an eigenvalue of $C$, prove $\frac{1}{\lambda}$ is an eigenvalue of $C^T$. I know $\lambda$ isn't zero because an orthogonal matrix has determinant $1$ ...
8
votes
3answers
562 views

The set of traces of orthogonal matrices is compact

Is the following set compact: $$M = \{ \operatorname{Tr}(A) : A \in M(n,\mathbb R) \text{ is orthogonal}\}$$ where $\operatorname{Tr}(A) $ denotes the trace of $A$? In order to be compact $M$ has to ...
0
votes
0answers
124 views

Orthogonality of stochastic matrix

Given a column stochastic matrix $P$, I wanted to give a relation between $\|P\|$ and orthogonality of $P$. One simple way to think about how close $P$ is to being orthogonal is $\|P^{\top}P - I\|$. ...
5
votes
1answer
662 views

Prove that the set of orthogonal matrices is compact

Let the set of all $n \times n$ matrices (denoted by $M_n(\mathbb R)$ ) be a metric space. Show that set of all orthogonal matrices is compact. My attempt: well i am beginner in real analysis. ...
17
votes
4answers
24k views

Why is the matrix product of 2 orthogonal matrices also an orthogonal matrix?

I've seen the statement "The matrix product of two orthogonal matrices is another orthogonal matrix. " on Wolfram's website but haven't seen any proof online as to why this is true. By orthogonal ...
3
votes
2answers
690 views

Is the trace set of orthogonal matrix compact?

Show that $$T = \{ \mbox{tr} (A) : A \in O_n (\mathbb{R}) \}$$ is compact. I tried to show this set is compact. I could not. Any hint would suffice. Thanks in advance.
4
votes
1answer
1k views

Determinant of identity minus product of orthogonal matrix and rank-$1$ matrix

I am interested in calculating, or bounding in some way, the following determinant \begin{equation} \det\left[\mathcal{I}-Rxx^t\right] \end{equation} Here, $Rxx^t$ is clearly a singular matrix. Im ...
22
votes
4answers
52k views

Eigenvalues in orthogonal matrices

Let $A \in M_n(\Bbb R)$. How can I prove, that 1) if $ \forall {b \in \Bbb R^n}, b^{t}Ab>0$, then all eigenvalues $>0$. 2) if $A$ is orthogonal, then all eigenvalues are equal to $-1$ or $...
17
votes
3answers
10k views

Why are orthogonal matrices generalizations of rotations and reflections?

I recently took linear algebra course, all that I learned about orthogonal matrix is that Q transposed is Q inverse, and therefore it has a nice computational property. Recently, to my surprise, I ...
4
votes
2answers
3k views

Center of the Orthogonal Group and Special Orthogonal Group

How can I prove that the center of $\operatorname{O}_n$ is $\pm I_n$? I have that $AM = MA$, $\forall M \in \operatorname{O}_n$ and $A^{-1} = A^T$, $M^{-1} = M^T$. Then $M = A^{-1}MA = A^{T}MA$. I ...
2
votes
2answers
7k views

Proof of orthogonal matrix property: $A^{-1} = A^t$

I have proofed this orthogonal property. Please correct it or show your version of the proof if I am wrong: $A^{-1} = A^t$ $A^{-1} \times A = A^t \times A$ $I = I$ I appreciate your answer
1
vote
1answer
405 views

Convex combination of orthogonal matrices

How would I show that the convex combination of orthogonal matrixes has spectral norm $ \leq 1$? (I have some idea how to do it ... but right now I'm stuck). Also, how would I prove that the unit ...
7
votes
2answers
4k views

Is sum of two orthogonal matrices singular?

I am trying to solve following problem. Let $A, B \in \mathbb{R}^{n\times n}$ be an orthogonal matrices and $\det(A) = -\det(B)$. How can it be proven that $A+B$ is singular? I could start with ...
10
votes
1answer
5k views

Orthogonal matrices form a compact set [duplicate]

Prove that the set of all $n \times n$ orthogonal matrices is a compact subset of $\mathbb{R}^{n^2}$. I don't know how it can be done. Thanks.
8
votes
1answer
1k views

Convex hull of orthogonal matrices

Where can I find the proof of the fact that the convex hull of the set of orthogonal matrices is the set of matrices with norm not greater than one? It is easy to show that a convex combination of ...
6
votes
1answer
2k views

constructive canonical form of orthogonal matrix

For every orthogonal matrix $Q$ over the reals there is an orthogonal matrix $P$ and a block diagonal matrix $D$ such that $D=PQP^{t}$. Each block in D is either $(1)$, $(-1)$ or a two dimensional ...
10
votes
2answers
4k views

Show that the set of all $n \times n$ orthogonal matrices, $O(n)$, is a compact subset of $\mbox{GL} (n,\mathbb R)$

I have only concept in topology, metric space, and functional analysis. How do I tackle this? Also I want to know that is the set connected?
20
votes
2answers
27k views

Difference between orthogonal and orthonormal matrices

Let $Q$ be an $N \times N$ unitary matrix (its columns are orthonormal). Since $Q$ is unitary, it would preserve the norm of any vector $X$, i.e., $\|QX\|^2 = \|X\|^2$. My confusion comes when the ...
6
votes
4answers
3k views

Is $O_n$ isomorphic to $SO_n \times \{\pm I\}$?

This question is taken directly from Artin's "Algebra", on page 150: Is $O_{n}$ isomorphic to the product group of $SO_{n} \times \{\pm1\}$? Here, $O_{n}$ is defined as the group of orthogonal ...
25
votes
2answers
50k views

In which cases is the inverse matrix equal to the transpose?

As said in the title, in which cases an invertible matrix is equal to the transpose? When is this: $ A^{-1} = A^{T} $ true? If the matrix A is orthogonal? Thank you!
1
vote
1answer
806 views

Completing a unitary matrix given a column

I am given a unit vector $e=1/\sqrt{n}(1,1,\ldots,1)'$ and the problem is to construct an $n \times n$ (real) unitary matrix $U$ which will contain $e$ as the last column. I understand that there are ...
14
votes
7answers
7k views

Why does $A^TA=I, \det A=1$ mean $A$ is a rotation matrix?

I know if $A^TA=I$, $A$ is an orthogonal matrix. Orthogonal matrices also contain two different types: if $\det A=1$, $A$ is a rotation matrix; if $\det A=-1$, $A$ is a reflection matrix. My question ...
4
votes
1answer
2k views

Orthogonal matrix over cross product

Let $a$ and $b$ be two unitary vectors in $\mathbb E^3$, and let $Q$ be an orthogonal matrix. Does the following hold? $$Qa \wedge Qb = \pm Q(a \wedge b)$$