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.

128 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
0
votes
0answers
349 views

Singular values uniquely determined for a matrix, does that mean $A=B$ IFF $\Sigma_A=\Sigma_B$?

I have heard it often that singular values for the SVD are "uniquely determined by the matrix $A$ in $A=U\Sigma V^*$ Wikipedia says it, and so do other sources. Now, I want to check how I am ...
0
votes
1answer
33 views

Simple linear regression and sum of squared errors

Let $Y_i$ be independent $N(\beta_0 + \beta_1x_i, \sigma^2$ for $1\leq i\leq n$, where $\{x_i\}_i=1^n, \beta_0, \beta_1, \sigma^2 >0$ are constants. Let $\hat{\beta_0}, \hat{\beta_1}$ be the ...
0
votes
0answers
21 views

Expressing an operator in fixed-angle bases

Fix $f$ an endomorphism of $\mathbb{R}^n$ and $\alpha \ne 0$. Suppose $B = \{ v_1, \dots, v_n \}$ is a basis of $\mathbb{R}^n$ with the following properties: $\langle v_i, v_j \rangle = \alpha$ for $...
0
votes
0answers
42 views

Multiplication of projectors

Let matrix $P_j$ denote the $m \times m$ orthogonal projector of rank $m-(j-1)$ that projects $\Re^m $ orthogonally onto the space orthogonal to span ($q_1 .......,q _{j-1}$), where $q_n= \frac {P_n ...
0
votes
0answers
158 views

If the condition number of a matrix is minimized then the matrix is orthogonalized?

I know that if a square matrix is orthogonal, then its condition number is 1. I was wondering if the converse is true ? I have a non-square structured matrix and I want that the matrix be orthogonal ...
0
votes
0answers
34 views

Which matrices generator $SO(p,q)$?

I know that $SO(2)$ is generated by the set of matrices of the form \begin{bmatrix} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{bmatrix} which can be used to find the ...
0
votes
0answers
37 views

how many orthogonal matrices are there in the residue field?

Could you please help me on this? Or just give me a hint how to start. Task: Calculate $\mid O_3(\mathbb{Z}/3\mathbb{Z})\mid$ and $\mid O_4(\mathbb{Z}/2\mathbb{Z})\mid$ where $O_n$ represents alle ...
0
votes
1answer
167 views

If vector, v = (1,4) then find the matrix of perp v

Can anyone please help me with this problem. A vector $\boldsymbol{v}$ is given with the coordinates $1, 4$ $\boldsymbol{v} = (1,4)$ Find the matrix of perp $\boldsymbol{v}$. As far as I know ...
0
votes
0answers
68 views

Conjugacy class in SO(4)

Is it true that a matrix of the form $$ R = \begin{pmatrix} \cos \theta_1 & \sin \theta_1 & 0 & 0\\ -\sin \theta_1 & \cos \theta_1 & 0 & 0\\ 0 & 0 & \cos \theta_2 & ...
0
votes
1answer
169 views

Computing the matrix derivative of $W^T T W$ w.r.t. W

I am trying to analytically and numerically compute the derivative of the following function $$ J(W) = \frac{1}{2}\|W^T R W - I\|_F^2 $$ From a paper that I am reading, the derivative of this ...
0
votes
1answer
92 views

If A is Orthogonal (mxm) matrix and (I+A) is invertible THEN Prove --> $ (I-A) (I+A)^{-1} $ is an Skew-Symmetric matrix

The doctor give us Question want to Prove that: If A is Orthogonal (mxm) matrix and (I+A) is invertible THEN Prove --> $$ (I-A) (I+A)^{-1} $$ is an Skew-Symmetric matrix My Question 1) how ...
0
votes
2answers
130 views

How to change the third column of this matrix so that it becomes orthogonal?

The matrix is as follows: $$ \begin{bmatrix} 3/5 & 4/5 & 3/5 \\ -4/5 & 3/5 & 0 \\ 0 & 0 & 4/5 \\ \end{bmatrix} $$ I get that in order for a matrix (call this matrix A) to be ...
0
votes
0answers
20 views

Orthogonal complement of a set

Could someone help me woth finding the orthogonal complement to the set \begin{bmatrix}1&1&0\\0&1&2\end{bmatrix} \begin{bmatrix}3&0&1\\1&2&1\end{bmatrix} (https://...
0
votes
0answers
22 views

Help understanding/proving : E(n) = O(n) ⋉ $\mathbb{R}^n$

I have been reading through some book, such as Geometry of Crystallographic groups (by Andrzej Szczepanski), and during this reading I came across the relationship: E(n) = O(n) ⋉ $\mathbb{R}^n$. ...
0
votes
0answers
52 views

Is there a relationship between a Householder reflection and one caused by subtracting unit vectors?

The original question: Let $\vec{q}$ be a unit vector $( \vec{q} \in \mathbb{R}^{n}, \left \| q \right \|=1 )$ and suppose that $\vec{q} \ne \vec{e}_{1}$ . Let $\vec{a} = \vec{q}-\vec{e}_{1}$ ​ and $...
0
votes
0answers
19 views

matrix law: $\sum_m a_{im}\,a_{jm}\,a_{km}\,a_{lm} >0$ for certain $(i,j,k,l)$

I search for a matrix with matrix elements $a_{im}$ that match the following rule: $$ \sum_m a_{im}\,a_{jm}\,a_{km}\,a_{lm} = \begin{cases} \alpha \quad\text{if } i=j=k=l \\ \beta \quad \text{if }i=j\...
0
votes
0answers
23 views

Findind the infimum of an isometry

It is well known that every isometry of $\mathbb{R}^n$ has the form $$x\to f(x)=Ax+a \,\, , $$ with $a\in\mathbb{R}^n$ and $A$ and $n\times n$ orthogonal matrix. How can we compute the infimum of the ...
0
votes
1answer
79 views

Orthogonal Projection onto a Sub-space

Can anyone help? I am really stuck on this question. Find the orthogonal projection of $(49 \ \ 49 \ \ 49 )^T$ on to the subspace $V$ of $R^3$ spanned by $(2 \ \ 3 \ \ 6)^T$ and $(3 \ -6 \ \ 2)^T ...
0
votes
1answer
666 views

Find the rotation/reflection angle for orthogonal matrix A

Just wanted to make sure my thinking is correct. The exercise key gives me a different answer than the one I'm getting and I'm not sure where I'm going wrong. Please don't give me the answer right ...
0
votes
1answer
117 views

An orthogonal matrix

Let $U$ be an orthogonal matrix with real spectra. Is it true that since the spectrum of $U$ is $\{-1,1\}$, so $U^{2}$ is similar to the identity matrix?
0
votes
0answers
36 views

Computation of $\det(I+UDV)$, $U,V$ unitary, $D$ diagonal

$I$ is the identity matrix, $U$ and V are both unitary matrices, $D $ is a diagonal matrix. Then how to get the determinant of $I+UDV$ efficiently?
0
votes
1answer
52 views

triangular/orthogonal matrix properties

Assume A and B are square orthogonal matrices, C and D are square upper triangular matrices with positive diagonals. A, B, C, D all have same dimensions (Q x Q). If AC=BD, then why does A=B and C=D?
0
votes
0answers
39 views

Orthogonal arrays

This is from a note I found I don't understand why t cannot equal to 3? If we choose first three columns, each row appears three times, for example, (0,0,0) appears three times in the subarray, which ...
0
votes
0answers
80 views

Isomorphism between $O(2n,\mathbb{R})$ and $O(n,n,\mathbb{R})$ and same question for their Lie algbera

Is there any isomorphism of Lie groups between $O(2n,\mathbb{R})$ and $G:=\{ X \in M_{2n\times 2n}(\mathbb{R}) \mid X^tSX = S\} $ where $S$=$\begin{pmatrix} & {I_n}\\ I_n & \end{pmatrix}$....
0
votes
0answers
55 views

Is a matrix that is orthogonally diagonalizable a projection matrix?

If a have a matrix say $A$ that is orthogonally diagonalizable (i.e. it can be written as $\lambda_1u_1u_1^T+ \lambda_2 u_2u_2^T+\dotsc \lambda_nu_nu_n^T$ , where the $u_i$ are the eigenvectors of the ...
0
votes
0answers
167 views

Is there a characterization of linear isomorphisms of the space of skew symmetric matrices?

Let $M_n^s$ denote the $\scriptstyle\binom{n}{2}$ dimensional space of $n \times n$ skew-symmetric matrices. Is there a characterization of linear isomorphisms that take $M_n^s$ into itself. If $n=2$ ...
0
votes
0answers
192 views

Finding an orthogonal matrix for a 3x2

I know how to find an orthogonal matrix for a $2\times2$ or $3\times3$ matrix. However I have been stuck on how to do this for a $3\times2$ matrix. The question is how to find a non-zero $3\times2$ ...
0
votes
0answers
126 views

Orthogonality of stochastic matrix

Given a column stochastic matrix $P$, I wanted to give a relation between $\|P\|$ and orthogonality of $P$. One simple way to think about how close $P$ is to being orthogonal is $\|P^{\top}P - I\|$. ...