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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Does orthogonal-invariance of a differential imply invariance of the function?

Let $U:\text{Hom}(\mathbb{R}^d,\mathbb{R}^d) \to \mathbb{R}$ be a smooth function . If $U$ is orthogonally-invariant, i.e. $U(QA)=U(A)$ for every $Q \in \text{SO}(n),A \in \text{Hom}(\mathbb{R}^d,\...
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Joint Gaussian PDF Change of Coordinates

My textbook says the following: Given a vector $\mathrm{\mathbf{x}}$ of random variables $x_i$ for $i = 1, \dots, N,$ with mean $\bar{\mathrm{\mathbf{x}}} = E[\mathrm{\mathbf{x}}]$, where $E[\cdot]$...
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Proving that a matrix is symmetric if it can be expressed as a spectral decomposition

If $\{u_1, \cdots, u_n\}$ is an orthonormal basis for $\mathbb{R}^n$, and if $A$ can be expressed as $$A = c_1u_1u_1^T + \cdots + c_nu_nu_n^T$$ then $A$ is symmetric and has eigenvalues $c_1, \...
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Consider the plane P in R-3 given by x-y-2z=0

I found the matrix A whose columns are a basis for P, A=[1,-1,-2] (vertical form). Using that I was able to find the projection matrix: P=$\frac{-1}{2} \left( \begin{array}{cc} 1 & -1 & 2 \\...
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Find constant $k$ in matrix so that matrix $A$ is orthogonal

Problem Find constant $k\in \mathbb{R}$ in matrix so that matrix $A$ is orthogonal when: $$ A = \begin{bmatrix} 1 & -1 & -7 \\ 1 & 3 & -1 \\ 2 & -1 & k \end{bmatrix} $$ ...
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47 views

Orthogonal Diagonalization of a $3$ by $3$ Matrix

$M$ $=$ $\begin{pmatrix}3&2&2\\ 2&3&2\\ 2&2&3\end{pmatrix}$. Diagonalize $M$ using an orthogonal matrix. So I got that the eigenvalues for $M$ were $1$ and $7$. For the ...
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74 views

Find out if a operator is self-adjoint or orthogonal

Let $V$ be a $\mathbb{R}$ inner product space, and $B=\left \{v_1, v_2, v_3 \right \}$ basis of $V$, with $||v_i||=1$ $\forall i=1,2,3$, and $<v_1, v_2>=<v_1, v_3>=0$ and $<v_2, v_3>=...
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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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108 views

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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82 views

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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2answers
56 views

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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3answers
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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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55 views

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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Prove matrix $A=(\cos\langle\alpha_i,\epsilon_j\rangle)_{n\times n}$ is orthogonal

Assume $\{\alpha_1,\cdots\alpha_n\},\{\epsilon_1,\cdots,\epsilon_n\} $ are both orthonormal basis of Euclidean Space $V$. Consider the matrix $$A=(\cos\langle\alpha_i,\epsilon_j\rangle)_{n\times n}$$ $...
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1answer
225 views

Nearest signed permutation matrix to a given matrix $A$

Let $A \in \mathbb{R}^{n\times n}$ be a square matrix and let $Q \in O(n)$ be the nearest orthogonal matrix to $A$ under the Frobenius norm, i.e. $$Q = \text{arg}\min_{M \in O(n)} ||A - M||_{F}^2$$ ...
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1answer
32 views

Project an orthogonal matrix onto the Birkhoff Polytope

It is known that the permutation matrices lie at the intersection of the orthogonal group $\mathbb{O}^N$ with the Birkhoff polytope $\mathbb{DS}^N$. It is also known that any non-negative matrix $X\in\...
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On a special decomposition of a $3\times 3$ matrix

Let $A\in\mathbb{R}^{3\times 3}$ be a diagonalizable matrix with strictly positive eigenvalues. (Note that $A$ is not required to be symmetric.) Let $A_S$ be the symmetric part of $A$, that is $$ A_S ...
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32 views

If vector $y_1 = (1,2,3)$ and $y_2 = (4,5,6)$, how to calculate orthogonal projector $\prod\{y_1\}$ onto subspace spaned by the vectors $y_1,y_2$?

If I have a vector $y_1 = (1,2,3)$ and $y_2 = (4,5,6)$, how to calculate $\prod\{y_{1}\}$ and $\prod\{(y_{i})_{1\leq i \leq2}\}$ according to the definition below? denote $\Pi \{ y_1,...,y_k\}$ ...
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if $Q^TQ = I$, can we get $QQ^T=I$

if we only know $Q^TQ = I$, can we get $QQ^T=I$? where $I$ is the identify matrix, $Q \in R^{m \times m}$
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$A^TA=B^TB$. Is $A=QB$ for some orthogonal $Q$?

Suppose that $A$ and $B$ are two real square matrices and $A^TA=B^TB$. Can we say that $A=QB$ for some orthogonal matrix $Q$? If they are vectors we have $\|a\|^2=a^Ta=b^Tb=\|b\|^2$, so intuitively ...
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Classical Lie group quotient-ed by its maximal parabolic subgroup

Let $B$ is a nondegenerate symmetric bilinear form on $\mathbb{C}^n$ then the corresponding complex orthogonal group is $\{g : GL(n, \mathbb{C}): B(gx, gy) =(x,y) \}$ In particular we use $$B (x,y) = ...
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75 views

Orthogonal Block Matrix

Given $A \in \mathbb{R^{\text{nxn}}}$, $B \in \mathbb{R^{n\text{x}m}}$ and $C \in \mathbb{R^{m\text{x}m}}$ such that $$ M = \begin{bmatrix} A & B \\ 0 & C \end{bmatrix} \in \mathbb{R^{...
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2answers
45 views

Do all orthogonal matrices satisfy both $Q^TQ = I$ and $QQ^T = I$?

Some definitions of orthogonal matrix (for instance, on Wikipedia) use the definition: $$ Q^TQ=QQ^T=I $$ While other definitions (for instance, on Wolfram MathWorld) only include one of the products:...
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66 views

If $V \in \mathbb{R}^{n \times n}$ is an orthogonal matrix, and $V^T e=0$, ($e$ is the ones vector) then is it true $V_m^T e = 0\, \forall \, m<n$? [closed]

If $V$ is an orthogonal matrix, and $V^T e=0$, then is it true that the sum of any subset of the columns also $= 0$? I am struggling to prove this, but I'm unable to come up with a counterexample.......
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$SO(N)$ generators to generate a basis for the space of $N\times N$ matrices

The generators of $SO(N)$ can be written as $(L_{ab})_{ij}=\delta_{ia}\delta_{jb}-\delta_{ja}\delta_{ib}$, with $1\leq i,j\leq N$ and $1\leq a<b\leq N$. Obviously, these generators for a basis for ...
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Transformation matrix for Higgs Doublet

If we consider two complex Higgs doublets with $SU(3)_{c} ⊗ SU(2)_L ⊗ U(1)_Y$ quantum numbers $\phi_{A,B} \sim (1,2,1)$, such that $$\begin{align}&\phi_A = \begin{pmatrix} \phi_{A}^{\dagger}\\ \...
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1answer
167 views

Do QR and RQ have the same eigenvalues [duplicate]

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-\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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2answers
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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 ...
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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 ...
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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 ...
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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{...
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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 ...
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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 ...
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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 ...
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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}...
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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 $3\times 3$...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $R^A_B$ ...
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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 (...
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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 $...
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1answer
53 views

Partitioning a rectangular matrix

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 ...