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.

# Questions tagged [orthogonal-matrices]

961 questions

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### What is the physical significance of the determinants of orthogonal matrices having the value of $\pm 1$?

I'm new to linear algebra and while studying orthogonal matrices, I found out that their determinant is always $\pm 1$. Why is that so? What could be the physical significance behind it?
I know that linear algebra can be intuitive when visualized,…

IntriguedMess

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### 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 pairs we…

Theodor

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### Why is inverse of orthogonal matrix is its transpose?

So the question is in the title. It's easy to prove when we know that there are real numbers in it and the dot product is standard. But why this works in the general case - when there are complex numbers inside and the dot product is something else?…

qiubit

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### 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 implication:
$\det(A)=-\det(B) \Rightarrow B$ is…

chip

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### 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?

Andy

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### 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.

emili

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### 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 orthogonal matrices has norm (I mean the norm as…

Emanuele Paolini

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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 $A$ (for 'nearest', use
the distance induced by…

ryanriess

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### Box-constrained orthogonal matrix

Given constants $\ell, u \in \mathbb{R}^{3 \times 3}$ and the following system of constraints in $P \in \mathbb{R}^{3 \times 3}$
$$
P^T P = I_{3 \times 3},\quad \ell_{ij} \leq P_{ij} \leq u_{ij},
$$
I would like to find a matrix $P$ which satisfies…

Alex Shtof

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### 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 matrices, $SO_{n}$ is the special group of orthogonal…

fourdriver01

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### 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 wish to know if it is correct.
So, I am going to…

Lelouch

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### 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 block of the form $\left(…

Oliver

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### Inequality for determinant of sum of two orthogonal matrices

My goal is to show that :
$$\forall A,B\in O_n(\mathbb{R}), |{\rm det}(A+B)| \le 2^n.$$
How can I access the determinant of the sum of two matrices ? I don’t know many ways to establish inequalities in Algebra, except with Bessel’s inequality and…

Julien

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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
$|b_{i,i}|>|a_{i,i}| \text{ for every } i \text{ and } |b_{i,j}|\leq |a_{i,j}| \text{ for every } i\neq j$.
I…

Lo Celso

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### How to prove that every orthogonal matrix has determinant $\pm1$ using limits (Strang 5.1.8)?

The problem of interest from Gilbert Strang's Introduction to Linear Algebra Section 5.1 is as follows.
Prove that every orthogonal matrix ($Q^TQ = I$) has determinant $1$ or $-1$.
(b) Use only the product rule. If $|\det(Q)| > 1$, then…

HiMatt

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