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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Which matrices can be realized as second derivatives of orthogonal paths?

$\newcommand{\skew}{\operatorname{skew}}$ $\newcommand{\sym}{\operatorname{sym}}$ $\newcommand{\SO}{\operatorname{SO}_n}$ I am interested to know which real matrices $A \in M_n$ can be realized as second derivatives of paths in $\text{SO}_n$…
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What does $A^{-1}=A^T$ have to do with “orthogonality”?

Whenever I read some use of the term “orthogonal”, I have been able to find some way in which it is at least metaphorically similar to the idea of two orthogonal lines in euclidean space. E.g. orthogonal random variables, etc. But I cannot see how…
user56834
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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 $$ \begin{pmatrix} \cos{\theta} & \sin{\theta}\\-\sin{\theta} & \cos{\theta} \end{pmatrix},$$ along with some diagonal…
Merkh
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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 be closed and bounded. $\|A\|=\sqrt {\sum_{i,j}…
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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. compact means every open cover has a finite subcover.…
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Can orthogonal matrix with positive diagonal have -1 in its spectrum?

Let $O\in \mathbb{R}^{n\times n}$ be an orthogonal matrix, i.e. $O^tO=I=OO^t$. Suppose its diagonal entries $\{O_{jj}\}_{j\in \{1,...,n\}}$ are (strictly) positive. Can $-1$ then be included in the spectrum of $O$? Note that if the diagonal is…
Thibaut Demaerel
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An open set invariant under a linear map implies it is an isometry or of finite order?

Let $U \subseteq \mathbb R^2$ be an open, bounded, connected subset. Let $A \in \text{SL}_2$ ($A$ is an invertible $2 \times 2$ matrix with determinant $1$) and suppose that $AU = U$. Must $A$ be either orthogonal or of finite order? If we assume…
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Does this orthogonal matrix parametrization have a name?

Given two linearly independent unit vectors $u, v \in \mathbb{R}^{n,1}$, the matrix $$ P(u, v) = I + 2 vu^{T}-\frac{(u+v)(u+v)^{T}}{1+u^Tv} $$ is the unique special orthogonal matrix that brings $u$ to coincide with $v$ and reduces to the identity…
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Nonnegative orthogonal matrices

Assume that $A \in \mathbb{R}^{n \times n}$ has nonnegative entries and $AA^T = I_n$ where $I_n$ is the identity matrix. Is it true that $A$ should be a permutation matrix? EDIT: I seem to have a proof for doubly stochastic matrices based on the…
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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.
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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 the desired (integer) eigenvalues. Generate an…
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Orthogonal Matrix with Determinant 1 is a Rotation Matrix

I am confused with how to show that an orthogonal matrix with determinant 1 must always be a rotation matrix. My approach to proving this was to take a general matrix $\begin{bmatrix}a&b \\c&d\end{bmatrix}$ and using the definition of a matrix being…
user258521
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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 see that since conjugating by $A$ must leave the…
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Why does gimbal lock occur “in two circles”?

Let $S^1=ℝ/2\pi ℤ$. Consider the Euler angle parametrization $$ T^3 \to SO(3),\\\ (ξ, υ, ζ)\mapsto R^Z_{ζ}R^Y_{υ}R^X_{ξ}, $$ where $T^3\cong S^1\times S^1\times S^1$ is the 3-torus and $$ R^X_ξ=\begin{pmatrix}1&0&0\\\ 0& \cos ξ&-\sin ξ\\\ 0& \sinξ…
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Let $M\in \text{SO}(3,\mathbb{R})$, prove that $\det(M-I_3)=0$.

Let $M\in \text{SO}(3,\mathbb{R})$, prove that $\det(M-I_3)=0$. My attempt: $$ \begin{align} \det(M-I_3)&=\det(M-M^TM)\\&=\det((I_3-M^T)M)\\&=\underbrace{\det(M)}_{=1}\det(I_3-M^T) \end{align} $$ Hence $$ \begin{align}…
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