# 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 ...