# 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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Is it true that a matrix of the form $$R = \begin{pmatrix} \cos \theta_1 & \sin \theta_1 & 0 & 0\\ -\sin \theta_1 & \cos \theta_1 & 0 & 0\\ 0 & 0 & \cos \theta_2 & ... 1answer 169 views ### Computing the matrix derivative of W^T T W w.r.t. W I am trying to analytically and numerically compute the derivative of the following function$$ J(W) = \frac{1}{2}\|W^T R W - I\|_F^2 $$From a paper that I am reading, the derivative of this ... 1answer 92 views ### If A is Orthogonal (mxm) matrix and (I+A) is invertible THEN Prove -->  (I-A) (I+A)^{-1}  is an Skew-Symmetric matrix The doctor give us Question want to Prove that: If A is Orthogonal (mxm) matrix and (I+A) is invertible THEN Prove -->$$ (I-A) (I+A)^{-1} $$is an Skew-Symmetric matrix My Question 1) how ... 2answers 130 views ### How to change the third column of this matrix so that it becomes orthogonal? The matrix is as follows:$$ \begin{bmatrix} 3/5 & 4/5 & 3/5 \\ -4/5 & 3/5 & 0 \\ 0 & 0 & 4/5 \\ \end{bmatrix} $$I get that in order for a matrix (call this matrix A) to be ... 0answers 20 views ### Orthogonal complement of a set Could someone help me woth finding the orthogonal complement to the set \begin{bmatrix}1&1&0\\0&1&2\end{bmatrix} \begin{bmatrix}3&0&1\\1&2&1\end{bmatrix} (https://... 0answers 22 views ### Help understanding/proving : E(n) = O(n) ⋉ \mathbb{R}^n I have been reading through some book, such as Geometry of Crystallographic groups (by Andrzej Szczepanski), and during this reading I came across the relationship: E(n) = O(n) ⋉ \mathbb{R}^n. ... 0answers 52 views ### Is there a relationship between a Householder reflection and one caused by subtracting unit vectors? The original question: Let \vec{q} be a unit vector ( \vec{q} \in \mathbb{R}^{n}, \left \| q \right \|=1 ) and suppose that \vec{q} \ne \vec{e}_{1} . Let \vec{a} = \vec{q}-\vec{e}_{1} ​ and ... 0answers 19 views ### matrix law: \sum_m a_{im}\,a_{jm}\,a_{km}\,a_{lm} >0 for certain (i,j,k,l) I search for a matrix with matrix elements a_{im} that match the following rule:$$ \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\...
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 ...