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.

128
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2answers
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What is the geometric interpretation of the transpose?

I can follow the definition of the transpose algebraically, i.e. as a reflection of a matrix across its diagonal, or in terms of dual spaces, but I lack any sort of geometric understanding of the ...
9
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1answer
2k views

Compactness of the set of $n \times n$ orthogonal matrices

Show that the set of all orthogonal matrices in the set of all $n \times n$ matrices endowed with any norm topology is compact.
14
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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 ...
7
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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 ...
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 $...
11
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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.
7
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2answers
395 views

How many $3 \times 3$ integer matrices are orthogonal?

Let $S$ be the set of $3 \times 3$ matrices $\rm A$ with integer entries such that $$\rm AA^{\top} = I_3$$ What is $|S|$ (cardinality of $S$)? The answer is supposed to be 48. Here is my proof and I ...
10
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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?
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 ...
9
votes
2answers
5k views

What are the properties of eigenvalues of permutation matrices?

Up till now, the only things I was able to come up/prove are the following properties: $\prod\lambda_i = \pm 1$ $ 0 \leq \sum \lambda_i \leq n$, where $n$ is the size of the matrix eigenvalues of the ...
19
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1answer
2k views

Why is the orthogonal group $\operatorname{O}(2n,\mathbb R)$ not the direct product of $\operatorname{SO}(2n, \mathbb R)$ and $\mathbb Z_2$?

We know that when $n$ is odd, $\operatorname{O}_n(\mathbb R) \simeq \operatorname{SO}_n (\mathbb R) \times \mathbb Z_2$. However, this seems not true when $n$ is even. But I have no idea how to prove ...
4
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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} &...
6
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3answers
2k views

Examples of matrices that are both skew-symmetric and orthogonal

Are there matrices that satisfy these two conditions? That is, a matrix $A$ such that $$A^T=A^{-1}=-A$$ What I know is that a skew-symmetric matrix with $n$ dimensions is singular when $n$ is odd.
4
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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 ...
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? $$Qa \wedge Qb = \pm Q(a \wedge b)$$
4
votes
1answer
149 views

Which matrices commute with $\operatorname{SO}_n$?

$\newcommand{\GLp}{\operatorname{GL}_n^+}$ $\newcommand{\SO}{\operatorname{SO}_n}$ Let $n>2$, and Let $A \in \GLp$ be an invertible real $n \times n$ matrix, which commutes with $\SO$. Is it true ...
1
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1answer
851 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 ...
1
vote
1answer
46 views

Let $q_1,q_2,q_3,q_4$ be orthonormal vectors in $\mathbb{R}^4$,$z_1,z_2,…,z_6$ orthonormal vectors in $\mathbb{R}^6$ and $A=z_1q_1^T + z_2q_2^T$.

Let $q_1,q_2,q_3,q_4$ be orthonormal vectors in $\mathbb{R}^4$,$z_1,z_2,...,z_6$ orthonormal vectors in $\mathbb{R}^6$ and $A=z_1q_1^T + z_2q_2^T$. a) Find the base and dimension of fundamental ...
20
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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 ...
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
6
votes
4answers
963 views

Generate integer matrices with integer eigenvalues

I want to generate $500$ random integer matrices with integer eigenvalues. Thanks to this post, I know how to generate a random matrix with whole eigenvalues: Generate a diagonal matrix $D$ with ...
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 ...
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 ...
3
votes
0answers
44 views

Is $Av_1,Av_2,Av_3$ orthogonal if you have eigenvector of $A^TA$

Let $A\in M_3(\mathbb R)$ and if $v_1,v_2,v_3$ orthonormed eigenvectors of matrix $A^TA$ and which eigenvalues is $1,2,3$ then vectors $Av_1,Av_2,Av_3$ is orthogonal? I only know that we need to ...
3
votes
1answer
168 views

On matrix tridiagonalization

Can any matrix $X \in \mathbb R^{n \times n}$ be decomposed into a tridiagonal matrix, i.e., $$X = P^{-1}DP$$ where $P \in \mbox{SO}(n)$ and $D$ is tridiagonal?
1
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4answers
3k views

Does $AA^T$ = I iff A is an orthogonal matrix?

I know that if $A$ is an orthogonal matrix, then $AA^T = I$. However, is it possible to have a non orthogonal square matrix but $AA^T = I$ as well? A square matrix of size $n$ is orthogonal if ...
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{...
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 ...
4
votes
1answer
212 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 ...
3
votes
2answers
73 views

Number of possible zero entries in orthogonal matrices

It's easy to check that in an orthogonal matrix $Q$ dimension $2 \times 2$ if there is entry $0$ in the matrix then necessary one additional zero must be present and the total number of zeros is $...
2
votes
1answer
763 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 ...
2
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0answers
259 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 ...
1
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1answer
502 views

When are matrix logarithms diagonal?

Is this true: "$P$ is diagonal if and only if there exists a diagonal $\log P$"? This is the matrix logarithm. I just want to make sure of my reasoning. If $\log P$ is diagonal, then $P$ is diagonal ...
1
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0answers
34 views

Does the following unitary matrix factorization have a name?

I know any unitary matrix can be factored as follows: $$\underline {\overline {\bf{U}} } = \left( {\prod\limits_{j = N}^1 {\underline {\overline {\bf{\Psi }} } \left( {{{\underline {\bf{w}} }_j}} \...
1
vote
3answers
76 views

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 ...
1
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3answers
467 views

How to prove the complement $P^\perp$ of a projection matrix $P$ have relation $I-P=P^\perp$

I want to know how to prove that for a projection matrix $P$ and its complement matrix $P^\perp$. We have $$I-P=P^\perp$$ I do know the intuition that $P$ and $P^\perp$ project a vector into two ...
0
votes
2answers
709 views

What is degrees of freedom of a real orthogonal marix?

How can I find the degrees of freedom of a $n \times n$ real orthogonal matrix? I have tried to proceed by principle of induction but I fail.Please tell me the right way to proceed. Thank you in ...
0
votes
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. ...