# 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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### Inequality for a positive matrix

I consider an $n\times n$ real positive-definite matrix $A$, which I rewrite in its diagonalized form: $A=O^TDO$, where $O$ is an orthogonal matrix and $D$ is diagonal. Since $A$ is positive-definite, ...
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### Smooth traversal of 𝑆𝑂(𝑛)

I am trying to constrain the space of matrices used for the layers of a neural network to those in 𝑆𝑂(𝑛). It is proven that 𝑆𝑂(𝑛) is a manifold. I'm trying to find a way to smoothly traverse ...
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### Which orthogonal rotation matrices for diagonalisation

We consider 2D orthogonal rotation matrices $R$. We consider a real matrix $$A = \begin{pmatrix} a & b\\ b & c \end{pmatrix}.$$ I write that $B = RAR^T$ for some diagonal matrix $B$. I would ...
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Question Let $A\in M_{n\times n}\left(\mathbb{R}\right)$ be a square symmetric matrix. Show that there exists constant $C>0$ such that for any $x\in\mathbb{R}^{n},x^{t}Ax\leq C\cdot\left\Vert x\... • 127 0 votes 1 answer 20 views ### Universal Properties of Orthogonal Matrices I wanted to ask this question since I have seen conflicting viewpoints on it. Are orthogonal matrices necessarily symmetric? I do not believe so but some website said they were so I need to confirm. (... 0 votes 0 answers 7 views ### Similarity relationship between orthogonal maps of the diagonal representation of$A + B$Question: What can be said in general of the relationships of diagonalization similarities between$A + B$and its summands? For example consider$f_{AB}(t) := t A + (1-t) B $a function that ... • 637 0 votes 1 answer 41 views ### Eigenvectors of an Orthogonal Matrix It is true that for orthogonal matrices, all eigenvectors of a single eigenvalue are orthogonal to all other vectors of different eigenvalues(distinct eigenbases are orthogonal). My question is are ... 0 votes 0 answers 27 views ### Decomposition of orthogonal complex matrix Note :$\mathrm{O}(n)$is$\mathrm{O}(n, \mathbb{R})$. I tried to solve the following exercise : For every complex orthogonal matrix$g \in \mathrm{O}(n, \mathbb{C})$, i.e.$g^{-1} = {}^\mathrm{t}g$, ... • 671 0 votes 1 answer 33 views ### How to understand the relationship of the fundamental subspaces in these big pictures? I am struggling like 6 hours to understand what this content in the middle mean. Can u get me some clue to interpret it? I understand all the stuff on the sides. So, row space is perp. to null space ... 1 vote 1 answer 28 views ### Spectral radius of a specific matrix multiplied by a diagonal matrix$D=diag(\mathrm{e}^{i\alpha_1},...\mathrm{e}^{i\alpha_n})$Let$A=(a_{ij})\in M_n(\mathbb C)$,$n\geq 3$be a matrix satisfying :$\sum_{j=1}^n\lvert a_{ij}\rvert^2=1,$for all$i=1,\ldots n$.$\sum_{i=1}^n\lvert a_{ij}\rvert^2\leq1,$for all$j=1,\ldots n$. ... • 121 0 votes 0 answers 30 views ### A matrix is orthogonal if its similar matrix is orthogonal I'm wondering how to prove this statement. Say$A=S^{-1}BS$and$A^TA=I$, then$(S^{-1}BS)^T S^{-1}BS = I$; so$ S^TB^T{S^{-1}}^{T} S^{-1}BS = I$, but I'm not sure about the next step. • 25 0 votes 1 answer 93 views ### How to prove: A geometric characterization of a$2 \times 2$orthogonal matrix$A$with$\mbox{det}(A) = 1$. In texts on linear algebra, a standard example for a$2 \times 2$orthogonal matrix is the rotation matrix given by $$R = \left[ \begin{array}{cc} \cos \theta & - \sin \theta \\ \sin \theta &... • 2,482 2 votes 0 answers 61 views ### Fast Multiplication of matrix and vector Multiplication of the DFT matrix and any vector can be implemented by FFT. I'm interesting about other fast Multiplication. Suppose there are three matrix of size N\times N, F , Q and P, where ... 0 votes 1 answer 33 views ### SVD of an orthogonal projector Here is my observation: Suppose there is an orthogonal projector P such that P=P^2. Then for arbitrary x, Px and (I-P)x are orthogonal. So we have$$ x^* P^* (I-P)x=0$$where A^* means ... • 619 3 votes 0 answers 34 views ### Gram Schmidt process against orthogonal basis W Another question that has a wrong answer from people adopting it. Am I wrong or the textbook wrong? Answer from book: My ans using the Gram-Schmidt process: such that \vec{x_1} and \vec{x_2} are ... • 643 2 votes 1 answer 44 views ### Understanding orthogonal projection of \vec{y} on to span of orthogonal set, with an example This is to verify if there's an issue with my understanding or if there's issue with the textbook. There seem to be also a previous question here on exactly the same, hoping to help myself and future ... • 643 3 votes 2 answers 126 views ### Differential form computation I was reading a text on differential geometry and I noticed this: "Given \omega=xdy\wedge dz+ydz\wedge dx+zdx\wedge dy\in\Omega^2(S^2) with S^2=\{p\in\mathbb{R}^3:\lvert p\rvert=1\}, show ... 0 votes 0 answers 44 views ### Matrix representation of a reflection Assume I have a vector v = (-1, 1, -1)^T and A is the relection through the plane orthogonal to v. How would I find the matrix representation of A? I have in my notes that the general ... 2 votes 0 answers 53 views ### Orthogonal Matrix with respect to different inner products From what I understand, an orthogonal matrix is one that satisfies A\cdot A^t = I_n. In such case, from what I saw online, \forall x\in \mathbb{R}^n,\;\left\Vert Ax \right\Vert =\left\Vert x \right\... • 271 0 votes 0 answers 22 views ### Conditions for orthogonality of Jacobi matrices Under which conditions is a Jacobi matrix of a coordinate transformation orthogonal? Background: I investigate one of the invariants of a rank two tensor under coordinate transformation with a ... • 21 1 vote 1 answer 80 views ### The conjugation action \mathbb{H}^*\times \mathbb{H} \rightarrow \mathbb{H} restricted to unit-quaternions yields an orthogonal representation Consider the action \mathbb{H}^*\times\mathbb{H} \rightarrow \mathbb{H}, (h,h')\mapsto hh'h^{-1}. Show that it preserves the orthogonal-decomposition \mathbb{R}\bigoplus Im\mathbb{H}, and ... • 341 1 vote 0 answers 67 views ### Question regarding the Three orbit case of classification of finite subgroups of SO(3). (Tetrahedral Group) When we have three orbit case, we would specifically have the subcase such that r_1,r_2,r_3=2,3,3 where each r_i represents the order of a stabilizer of some pole p_i. If we let the finite ... • 1,248 2 votes 1 answer 42 views ### Curve with a prescribed Frenet frame Suppose you are given an antisymmetric matrix X. Then A(t)=e^{Xt} is a curve of orthonormal matrices, with A(0)=Id. Is it possible to construct a curve whose Frenet frame vectors T,N,B are the ... • 13.2k 2 votes 0 answers 64 views ### Questions regarding a proof of "classification of the finite subgroups of SO(3)". I'm self-studying the classification of finite subgroups of SO(3) with this paper. On page 13 of the paper, there is a paragraph (below the equation (9.3)) that I'm having trouble understanding. The ... • 1,248 1 vote 0 answers 126 views ### Is there a name for this 'splitting' of an orthogonal structure into unitary and symplectic structures? Suppose we have a (non-trivial) representation of some special orthogonal group SO(p,q) over a real vector space V, I.e. the action of elements of SO leave invariant a non-degenerate symmetric ... • 434 0 votes 0 answers 30 views ### Commutativity of Multiplying Semi-orthogonal Matrix with Symmetric Matrix I encounter the following question when I read the paper "Quadratic Optimization with Orthogonality Constraints" (https://arxiv.org/pdf/1510.01025.pdf). Simply speaking, let X\in \mathbb{R}^... 1 vote 1 answer 40 views ### QR decomposition and a change of basis Consider a square matrix A and its QR decomposition A = QR, where Q is orthogonal and R is upper triangular matrix. Now consider a change of basis. Let A' = PAP^T, where P is an ... • 11 0 votes 0 answers 52 views ### If a linear map sends orthonormal basis on orthonotmal basis then it is an isometry? Let (\mathbf{u_1,u_2,u_3}) and (\mathbf{v_1,v_2,v_3}) orthonormal lists. Define T:\mathbb{R^3}\rightarrow \mathbb{R^3} through the lineal extension T(u_k)=v_k. Is T an isometry? My attempt: I ... • 301 1 vote 0 answers 26 views ### Quadratic forms and orthogonal basis I have the quadratic form q(x,y,z) = x^2 - 2y^2 +xz +yz Give the polar form and the matrix of q in canonic basis. I think this question is ok :$$ A = \begin{pmatrix} 1 & 0 & 1/2 \\ 0 &... • 21 2 votes 1 answer 72 views ### When$A \in SO(3)$,$A$is always a rotation. I have official proof of this problem, but I am having trouble understanding some parts of it. Thus, I would like to share the parts and would like to get checked if my understanding is correct. ... • 1,248 0 votes 2 answers 222 views ### Number of independent elements of an orthogonal matrix I don't understand the following fact. The$n^2$elements of an orthogonal matrix$A$of order$n \times n$are not independent. This follows from the fact that$A′A = I_{n}$which implies that the ... • 173 2 votes 1 answer 69 views ### How to show that there is a non-zero$v \in \mathbb{R}^n$for any$A \in O(n)$s.t$Av=\pm v$whenever$n$is odd. Intuitively, this makes sense, but just in$\mathbb{R}^3$. For example, in$\mathbb{R}^3$, when any$A$is given, this$A$determines how much should a point in$\mathbb{R}^3$be rotated about the ... • 1,248 0 votes 0 answers 63 views ### Decomposition of a matrix with orthogonal columns Note that this question is also posted in mathoverflow. I'm new to the community so I'm also posting this here if it is more appropriate. I am given a square matrix$A = [A_1, A_2, ..., A_n]$, where ... • 1 0 votes 0 answers 34 views ### Covariance of entries of a random orthogonal matrix Let$Q$be an$n\times n$matrix that follows the Haar measure (that is, we begin with an$n\times n$matrix of standard normal random variables and perform Gram-Schmidt orthogonalization) I am trying ... • 986 2 votes 1 answer 54 views ### Confusion about connection between orthogonal matrices and rotation in higher dimensions I found this question which discusses that all orthogonal matrices are rotations/reflections, since the map$X\rightarrow AX$preserves the scalar product, with$A\$ an orthogonal matrix (see proof in ...
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Suppose I $$\begin{array}{ll} \underset{\Omega \in \Bbb R^{n \times n}}{\text{maximize}} & \text{Tr}(\Omega A)\\ \text{subject to} & \Omega^\top \Omega = I_n\end{array}$$ What am I looking at ...