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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How to show that $SO(3)$ and $\mathbb{R}P^3$ are dffeomorphic?

Pardon me for repeating a question, but I am not able to justify one aspect of this proof that $SO(3)$ and $\mathbb{R}P^3$ are diffeomorphic, and I was hoping that someone could clarify it for me. ...
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Show that Y is isotropic in V iif, for some v in V, the random vectors Y-v and O(Y-v) have the same mean and variance

Problem: Show that Y is isotropic in V iif, for some v in V, the random vectors Y-v and O(Y-v) have the same mean and variance for all orthogonal linear transformations O: V --> V I understand that ...
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Can we write any unitary matrix as the exponential of a skew-symmetric complex matrix?

Following the article https://en.wikipedia.org/wiki/Skew-symmetric_matrix#Infinitesimal_rotations which states that a skew symmetric (or anti-symmetric) real matrix can be written as the exponential ...
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Proof of orthogonal and symmetric.

Given $x$ is an $n$ dimensional vector, if $A = I_n- (2/x^Tx)xx^T$, show that it is orthogonal and symmetric. I know that if $A$ is orthogonal and symmetric, $A = \operatorname{inverse}(A) = A^T$, ...
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Is the solution to $A-O(A)=\tilde \Sigma$ unique?

Let $\tilde \Sigma=\text{diag}(\tilde \sigma_i)$ be a diagonal matrix, with $\tilde \sigma_i>0$. ($1 \le i \le n$). Suppose that $A$ is a real invertible $n \times n$ matrix with positive ...
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Linear-algebra first course problem about orthogonal matrices

I am trying to demonstrate next assert about matrices: $A$ is a matrix of $n$ order, with $n$ odd, that obeys $A A^T =I$ and $\det\, A=1$. Then $\det\,(A-I)=0$. I have tried a number of things but ...
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Equivalence classes of orthogonal matrices

Consider $O(n)$, the Lie group of $n{\times}n$ orthogonal matrices. Let $\pi$ be a given permutation of $(1,2,\ldots,n)$; the elements of the corresponding permutation matrix are \begin{eqnarray} P_{...
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Let $A=QR$ where $Q\in$ $M_n(\mathbb R)$ orthogonal and $R\in M_n(\mathbb R)$ is upper triangular matrix. Do $A$ and $RQ$ have the same eigenvalues? $\det(A-\lambda I)=\det(QR-\lambda I)=\det(QR-\... 2answers 82 views If matrix$A$has orthogonal columns, then what is solution of$Ax=b$Let$A\in M^4(\mathbb R)$is matrix which column is$a_1,a_2,a_3,a_4$they are orthogonal and length is$2,1,3,2$. a) Write an explicit form for the solution of$Ax=b$b) Write matrix as sum of four ... 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 ... 2answers 37 views $3\times 3$rotation matrix from axis of rotation and Angle I had encountered problem that tells to find$3\times 3$rotation matrix from axis of rotation$(1,1,2)$and Angle$\pi /3$. I know that for axis of rotation on some standard vector like x axis, By ... 1answer 84 views Can we characterize the equivalence classes of matrices up to left multiplication by an orthogonal matrix? Let$M_n$be the space of$n \times n$real matrices, and consider the following equivalence relation on$M_n$:$A \sim B$if there exist$Q \in O(n)$such that$A=QB$. Can we characterise nicely ... 1answer 45 views Characteristic polynomial roots of$ 3\times 3$orthogonal matrix. How would you approach this problem? Let$A$be an orthogonal$3$by$3$matrix. That is,$A^TA = AA^T=I_3$. Prove that the characteristic polynomial$\textit{p}_A$has a real root. I am not ... 2answers 42 views Reflection matrix - are these two definitions equivalent? Are these two definitions of a reflection matrix Q equivalent (if and only if)? Definition 1:$Q^TQ = I$and$det(Q) = -1$Definition 2:$Q = I-2nn^T$where$n$is a unit normal vector to the ... 1answer 52 views If the product of two orthogonal matrices is diagonal, is there a relation between the matrices? Suppose an SVD decomposition of a$3\times3$real invertible matrix$F = U K W$, where$ U U^T = U^TU=WW^T=W^TW=I$and$K = $diag$(k_1,k_2,k_3)$. Now suppose$ F = A B $where$A$and$B$have SVD ... 1answer 247 views Properties of orthogonal matrix Let$u$and$v$be orthogonal 3d unit vectors. Let$w = u \times v$and $$A=\begin{bmatrix} u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3\end{bmatrix}$$ Which of ... 0answers 37 views Name of orthogonal/unitary matrix decomposition? Suppose we have an orthogonal/unitary matrix$T$of even dimension. Then we can decompose it into: $$T = \begin{bmatrix} U_1 & 0 \\ 0 & U_2 \end{bmatrix} \begin{bmatrix} R & -\sqrt{... 6answers 923 views What is the physical significance of the determinants of orthogonal matrices having the value of \pm 1? I'm new to linear algebra and while studying orthogonal matrices, I found out that their determinant is always \pm 1. Why is that so? What could be the physical significance behind it? I know that ... 1answer 71 views Spheres and orthogonal matrices as spaces of solutions to matrix equations For a \in \mathbb{R}^n, the solutions to$$1=\sum_{i=1}^na_i^2$$form an (n-1)-sphere in \mathbb{R}^n. Meanwhile, for A \in \mbox{GL}_k (\mathbb{R}), the solutions to$$1=AA^T$$are the ... 0answers 25 views Find a mapping from R^{n \times n} to orthogonal matrices in the unit sphere such that distances do not increase. Let O be the orthogonal group, let B = \{A \in R^{n \times n} \text{ | } \lVert A \lVert \leq 1\}. Find a function F:R^{n \times n} \rightarrow O \cap B, such that \forall_{A \in R^{n \times ... 5answers 328 views Determining the last column so that the resulting matrix is an orthogonal matrix Determine the last column so that the resulting matrix is an orthogonal matrix$$\begin{bmatrix} \dfrac{1}{\sqrt{2}} & \dfrac{1}{\sqrt{6}} & ? \\ \dfrac{1}{\sqrt{2}} & -\dfrac{1}{\sqrt{6}... 2answers 63 views If$\sigma:M_3\rightarrow S_3$is a linear map such that$\sigma(PMP^{-1})=P\sigma(M)P^{-1}$, then$\sigma(M)=\ldots$In page 32 of the book A Mathematical Introduction to Fluid Mechanics, by Alexandre Chorin and Jerrold E. Marsden, it is used and commented the following property: let$M_3$be the space of$3\times 3$... 1answer 23 views Transforming a projection matrix Suppose we have an orthogonal projection$P$of rank$r$, which has an eigendecomposition$P=Q\begin{bmatrix}I_r & 0 \\ 0 & 0\end{bmatrix}Q^T$. Here$Q$is an orthogonal matrix. Is it true ... 1answer 35 views Prove: if A is a transformation matrix of orthonormal bases then A is unitary I'm stuck proving this statement, perhaps because too many indexes are involved.$A \in M \tiny nxn$(F) Prove that if there exist B', B orthonormal bases of$F^n$, such that A is a transformation ... 3answers 408 views 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 ... 0answers 67 views Orthogonal complement of a vector I have a matrix (in this case it's a vector but it could be a matrix)$\alpha$. $$\alpha'=(-0.086877710\quad03767617)$$ In the paper describing the methodology I try to implement, this definition is ... 0answers 93 views Equivalence of Alias and Alibi interpretation of Rotation Matrices. Let's consider two right handed frames$\mathcal R_A$, with coordinate axes$i_A$,$j_A$and$k_A$and$\mathcal R_B$, with coordinate axes$i_B$,$j_B$and$k_B$. Defining the rotation matrix$R^A_B$... 6answers 608 views Prove that$-1$is an eigenvalue of an orthogonal matrix$A \in M_{4 \times 4} (\Bbb R)$with$\det(A)=-1$[duplicate] Let$A \in M_{4 \times 4} (\Bbb R)$be an orthogonal matrix with$\det(A)=-1$. Prove that$-1$is an eigenvalue of$A$. I'm a bit lost. I know about all the basic orthogonal matrices' properties (... 1answer 54 views A particular subset of row-orthogonal matrices Let$O\in\mathbb{R}^{n\times m}$,$m>n$, be a matrix with orthonormal rows, that is$O O^\top =I_n$, where$\bullet^\top$denotes transposition and$I_n$the$n\times n$identity matrix. Partition$...
We are given an $m\times n$ matrix $A$, where $m>n$, and a symmetric orthogonal projection matrix $P$. The projection matrix has $p$ nonzero eigenvalues and $p<n$. Is it possible to find a ...