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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30 views

Is spectral theorem an equivalence?

A matrix $A \in K^{n \times n}$ is diagonalizable with orthonormal basis on $K$ if there exists $P \in K^{n \times n}$ orthogonal such that $P^*AP = D$ where $D \in K^{n \times n}$ is a diagonal ...
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Invertibility of Similar and Orthogonal Matrices [closed]

Are matrices similar to orthogonal matrices invertible?
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Is the orthogonal basis obtained for $A$ a basis for $B^{\bot}$ where $B$ is a subspace of $A$?

Let $A$ be an inner product space of dimension $n$ with inner product $\langle\,,\,\rangle$. Let $B$ be subspace of $A$, and let $\{a_1, ..., a_m\}$ be a basis for $B$. Suppose that $\{a_1, ..., a_m, ...
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Understanding vector product invariance under orthogonal transformation

I am reading Do Carmo's "Differential Geometry of Curves and Surfaces", and there is an exercise in it that asks us to prove that the cross product (vector product) is invariant under ...
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Generators of the orthogonal group

The real orthogonal group $O(N) = \{ A \in GL(N,\mathbb{R})|A^TA = AA^T = 1 \}$ consists of rotations with $\det A = +1$ (forming the subgroup $SO(N)$) and of reflections with $\det A = -1$. Its ...
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Relation between $SU(2)$ and $O(3)$

This is almost identical (even one may say "duplicate" of) to this relation between the group $O(3)$ and $SU(2)$. But i would like to ask one more question. Does this relations (given at the ...
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Find a “common” row space and column space for a matrix, solving $\max_{P\in \mathbb O^{n\times k}} \quad \|P^TB P\|_F^2$ given $B$

I have an optimization problem as follows: $$\min_{\substack{A\in \mathbb R^{k\times k}\\P\in \mathbb O^{n\times k}}}\|B - P \cdot A\cdot P^T \|_F^2$$ where $B$ (whose rank can be thought as greater ...
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show that $M \mapsto \max( \{ \operatorname{Tr}(OM) \mid O \in O_n ( \mathbb R )\})$ with is continuous

\begin{align} f : M_n (\mathbb R) & \to \mathbb R \\ M & \mapsto \max(\{\operatorname{Tr}(OM) \mid O \in O_n ( \mathbb R )\}) \end{align} prove that $f$ is well defined and continuous Let $M\...
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Finding the spectral decomposition of a given $3\times 3$ matrix

I am trying to find the spectral decomposition of the following matrix: $$ A=\begin{pmatrix} 4 & 0 & 0\\ 0 & 2 & 0\\ 2 & 3 & 0\\ \end{pmatrix} $$ So I found the eigenvalues ...
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Show proofs for inverse of these singular matrix.

I tried really hard, but I have no idea how to approach this question. A and B matrix are not invertible, so inverse does not exist. So, how do I go about proving them ? Simply saying they do not have ...
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$A$ is positive semidefinite $\iff \text{det} (B_K) \geq 0$

Let $A \in \mathbb R^{n \times n}$ a symmetric matrix. Show that $A$ is positive semidefinite $\iff$ all its symmetric minors are $\geq 0$, that means $\det(B_K) \geq 0$ for all $K \subseteq \{1,\...
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Orthogonal transformation of a set of points to positive orthant

Let $x_1,x_2\ldots,x_n$ are vectors from $\mathbb{R}^d$. Also assume that $x_i^{\top}x_j \geq 0$ for all $i,j=1,2,\ldots,n$. I am wondering if there is an orthogonal matrix $W$ such that the entries ...
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Scalar multiples of orthogonal matrices

For orthogonal matrices, I know that $A^TA = AA^T = I$. But if we have some matrix $AA^T = \lambda I$, where $\lambda$ is an integer scalar, I believe it's still true that $A^TA = AA^T = \lambda I$. ...
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Why the product of at most $n(n-1)/2$ Givens rotations can represent a rotation matrix?

As title. Real orthogonal matrix with determinant 1 is an rotation matrix, right? I saw the saying like in another question or this paper, but it seems everyone just claim so. Why the upper bound is $...
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Let $X, Y ∈ R^{n × m}$. We are looking for an orthogonal matrix $R ∈ M_n (R)$ that minimizes $\|RX - Y\|^2=\operatorname{tr}((RX - Y)^T (RX - Y))$.

I'm trying to show that the wanted matrix is $R = VU^T$, where $XY^T = UΣV^T$ is the singular value decomposition, but I'm having trouble understanding what it means or how to start.
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Find the matrix of the rotation matrix $U$

The matrix of $U$ in $\mathbb{R^3}$,with the standard inner product which is rotation of the plane $W=sp\{\alpha_1,\alpha_2\}$ about the orthogonal line $\alpha_3$ through the angle $\theta$ , where $\...
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Why is $Tr(AP)=\sum^p_{j=1}\lambda_j(e_je_j^TBB^T)?$

Let $P$ be a $p\times p$ orthogonal projection matrix, $A$ be a $p\times p$ symmetric matrix. I've been going through a solution to an exercise in my book, and I was confused as to why would $$Tr(AP)=\...
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Is it possible to find the orthogonal matrix, $Q$, in $QMQ^\text{T} = A$?

Is there a way to solve for the orthogonal matrix, $Q$, in $QMQ^\text{T} = L^\text{T}GL$ where $M$ is a known anti-diagonal matrix and the right hand side is known ($L$ is lower-triangular from a ...
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Finding vectors in orthogonal complements to create a unique sum

Take U,W to be subspaces of $\mathbb{R}^{3}$ $U = \operatorname{Lin}\left\{\left(\begin{array}{c} -1 \\ 0 \\ 1 \end{array}\right),\left(\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right)\right\}$ and $V=\...
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Compute $[P]_{\beta}$, $\beta$ canonical basis for $\mathbb{R}^{3}$ and $P: \mathbb{R}^{3} \to \mathbb{R}^{3}$ is the orthogonal projection operator.

I stumbled upon this problem in a list of homework extra exercises. Let $P: \mathbb{R}^{3} \to \mathbb{R}^{3}$ be the orthogonal projection operator of $\mathbb{R}^{3}$ over the plane $x_{1} +2x_{2}-...
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Is the rank of a linear operator over symmetric matrices preserved after rotating its domain?

I'm having a hard time trying to prove this, but somehow it seems true... The problem is: Let $\mathbb{S}^m$ be the space of all real symmetric matrices, and let $U\in \mathbb{R}^{m\times m}$ be an ...
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Showing orthogonal matrix for $\left\langle {u,v} \right\rangle = \left\langle {Au,Av} \right\rangle$

Show the $A$ is a orthogonal matrix when the $A$ satisfying $\left\langle {u,v} \right\rangle = \left\langle {Au,Av} \right\rangle$, $\forall u, v \in \mathbb{R}^3$ (The $A$ is a $3 \times3$ matrix.) ...
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The matrix of an isometry has orthonormal columns

Axler's Linear Algebra Done Right proves that if $T : V \to V$ is a linear operator on a finite-dimensional inner product space over $F \in \{ \mathbb{R}, \mathbb{C} \}$, then the following are ...
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Block diagonalization of an orthogonal matrix

Consider an orthogonal matrix $A\in \mathbb{R}^{3\times 3}$. Find an orthogonal matrix $T\in O(3)$ s.t. \begin{equation} T^\top A T=\begin{pmatrix}1&0&0\\0&\cos\theta&-\sin\theta\\0&...
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Question about finding an orthogonal matrix of an orthogonal matrix which can be written as $C^{tr}MC$

Sorry guys, I have a problem about finding an orthogonal matrix $C \in Mat_{nxn}(\mathbf R)$ such that $$ C^{tr}MC= \begin{bmatrix} \lambda &0&0\\ 0 &cos(\alpha)&sin(\alpha)\\ 0 &-...
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Is orthogonal matrix diagonalizable?

My understanding was that every normal matrix is diagonalizable. However I recently read that this applies to unitary but not necessary to orthogonal matrices. "For every orthogonal operator of ...
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$3×3$ orthogonal matrix $T$ fixes 2 points in unit sphere

Let $T:\mathbb{R^3} \rightarrow \mathbb{R^3}$ be an orthogonal transformation such that $\det T = 1$ and $T\neq I$. Let S be the unit sphere in $\mathbb{R^3}$. I need to show that $T$ fixes exactly ...
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Diagonalization of a real symmetric matrix

I saw in the spectral theorem that any real symmetric matrix $A$ is diagonalizable by orthogonal matrices; i.e., given a real symmetric matrix $A$, $Q^{T} AQ$ is diagonal for some orthogonal matrix $Q$...
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If a square matrix's column vectors and row vectors all have norm 1, then is the matrix orthonormal?

Well, this is a question I had to ask myself while solving a problem that asked me to prove a matrix is orthonormal. I could show that both the column vectors and the row vectors of said matrix all ...
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Is there a natural way to “project” an arbitrary matrix to an orthogonal matrix?

I am dealing with an optimization problem where I need to find an optimal rotation matrix. Let me first formulate the problem. Input: An initial rotation matrix $M\in SO(3)\subset\mathbb{R}^{3\times ...
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Saddle Points and Local Maxima on an Orthogonal Tensor Maximizing function

For orthonormal vectors $v_i$ consider the following orthogonal tensor: $$T = \sum_{i=1}^n v_i \oplus v_i \oplus v_i \oplus v_i $$ and the maximizing function: $$T(x, x, x, x) - ||x||^6 = \sum_{i=1}^n ...
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$QR$ decomposition of block matrix

Given a square block matrix $$ M = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \in \mathbb R^{2d \times 2d}, $$ where $A, B, C, D \in \mathbb R^{d \times d}$. Is there some kind of a block $...
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How can I obtain $B = B^{T}$ if $A = 2 B − I$ is an isometry and $B^2 = B$?

Prove that matrix $A=2B−I$ is an isometry (where $A^{2}=I$) $\iff$ $B$ is an orthogonal projector, i.e., $B^2=B=B^{T}$. For now, I've just proved that $B=B^2$ using $A=2B−I$. But I have no idea how I ...
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Show that $(A-I)^{-1}(A+I)$ is orthogonal if $A$ is skew-symmetric

In this question, I saw the proof, but I don't get it. $$\begin{aligned} (A - I)^{-1}(A + I) \left( (A - I)^{-1}(A + I) \right)^T &= (A - I)^{-1} (A + I) (A^T + I)(A^T - I)^{-1} \\ &= (A - I)^{...
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Show that $O_2(\mathbb{R})$ contains only rotational and reflective symmetries.

Show that $O_2(\mathbb{R})$ contains only rotational and reflective matrices. I know that rotational and reflective symmetries are part of $O_2(\mathbb{R})$. I want to show that there is no other ...
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Definition of orthogonal projection of a matrix

Given an $m$ x $k$ orthogonal matrix $Q$ and an $m$ x $m$ matrix A, I know that the matrix $QQ^{T}A$ is the orthogonal projection of $A$ onto the column space of $Q$. Recently, in a paper, I read that ...
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$SU(2)$ and the $3$-sphere, with $SO(2)$ a longitude

I'm currently studying the connection between $SU(2)$ and the $3$-sphere. A longitude on the sphere is of the form $QTQ^*$ where $Q \in SU(2)$ and $T = \{D_\lambda : \lambda \bar{\lambda} = 1 \} $, ...
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Cardinal of the Special Orthogonal Group $SO_n\left(\mathbb{Z}/2^n\mathbb{Z}\right)$

Let $SO_n(R)$ be the group of matrices with coefficients in a ring $R$ whose determinant is equal to $1$. I remembered an old question asked at an oral exam : $$\text{What is the order of the group }...
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What happens to the trace if you multiply with orthogonal matrices

Suppose we are given matrices $A$ and $Q$. Furthermore denote by $U$ and $V$ orthogonal matrices. For a matrix $Q'$ the equality $Q'=UQV^T$ holds. Since $U$ and $V$ are orthogonal matrices, the ...
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If $f: (-1,1)→M_n(\mathbb R)$ be a $C^1$ map such that $f(0)=I$ and $f(t)\in O(n)$ ,show that $Df(0)$ exists and is a skew-symmetric matrix

If $f: (-1,1)→M_n(\mathbb R)$ be a $C^1$ map such that $f(0)=I$ and $f(t)\in O(n)$ ,show that $Df(0)$ exists is skew-symmetric . matrix Actually, we have $f(t).f(t)^T=I$ for all $t,$ and we are to ...
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On state-transition matrix and fundamental theorem of local theory of curves

Lately I was faced by a problem and I can't think some way to solve it: Let $A(t)$ be an antisymmetryc matrix, for all $t \in I$. Prove that $\phi(t,t_{0})$ (the state-transition matrix) is an ...
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Dimensions of Orthogonal Group Representations

I'm aware that the irreducible representations of the orthogonal group $O(n;\mathbb{C})$ are labeled by partitions $\lambda$ such that the sum of the first two columns of $\lambda$ is at most $n$. Is ...
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Orthogonal transformations and matrices having similar eigenvalues

If there is an orthogonal similarity between symmetric matrices $A$ and $B$ by having $B=OAO'$ for an orthogonal matrix $O$ ($'$ is transpose) we infer $A$ and $B$ are having identical eigenvalues (...
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An orthogonal matrix the given power of which is the identity matrix

What is the general form of a real orthogonal matrix of $n$th order that satisfies the condition $A^p=E$ where $p$ is a given positive integer, and $E$ is the identity matrix? As I see, in the case of ...
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Solving for $U$,$\Sigma$ and $V$ in SVD of a matrix

Knowing from theory that fro a matrix $A$ we $A = U\Sigma V^{T}$ I want to solve for $U$,$V$ and $\Sigma$. My effort is the following but I don't if I am correct. \begin{align*} A &= U\Sigma V^{T} ...
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Intuitive proof for $(a, b) = (a', b')$.

If $p,q,r,s$ are integers such that $ps - qr = \pm 1$, and $a,b,a',b' $ are integers such that $a' = pa +qb, b' = ra + sb$, prove that $(a, b) = (a', b')$. Here, the particular value of $ps - qr = \pm ...
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117 views

Commutation of symmetric and skew-symmetric part of orthogonal matrix

Can we claim that for an orthogonal matrix $A$ (satysfying $AA^T=I$), its symmetric and skew-symetric parts are always commuting? Symmetric part is calculated as $S=\frac{1}{2}(A+A^T)$ and skew ...
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78 views

How to find this integral over an annulus

i have $v=\sum_{g\in G_k} g v_r\in W^{1,N}_{0,G_k}(\Omega_r)\setminus\{0\}$. $$\Omega_r=\{x\in \mathbb{R}^N, r<|x|<r+1\}, r>0, N\geq 2, N\neq 3$$ $$O_k=\{g\in O(2): g(x)=\left(x_1 \cos\frac{2\...
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52 views

Is an orthogonal matrix with determinant $-1$ a rotation matrix? [closed]

I know that an orthogonal matrix with determinant $1$ is a rotation matrix. However, is an orthogonal matrix with determinant $-1$ also a rotation matrix?

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