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 can I find the general form of an orthogonal matrix?

I know that the general form of orthogonal matrices is $$\begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix}$$ since they are all rotation matrices but how do ...
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If the matrix representation of an orthogonal transformation with respect to a basis is an orthogonal matrix, the basis is an orthonormal basis?

I'm learning linear algebra and interested in the relationship between linear transformation, matrix representation and the basis. The followings are my questions: If the matrix representation of an ...
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Projection onto the Set of Orthogonal Matrices - $\mathcal{O}^{n} = \left\{ X \in \mathbb{R}^{n \times n} \mid {X}^{T} X = I \right\}$

Let $\mathcal{O}^{n}$ be the set defined by $\mathcal{O}^{n} = \left\{ X \in \mathbb{R}^{n \times n} \mid {X}^{T} X = I \right\}$, namely the set of Orthognal Matrices of size $n \times n$. I ...
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Matrix inversion problem in ridge regression

The ridge estimator can be written in the following way, where the singular value decomposition of X is $X=UDV^{'}$. I can't quite figure out how the last step (4th step) was obtained from the 3rd ...
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SVD of orthogonal matrix and multiplying the left and right singular matrices?

I came across some code that takes some orthogonal matrix $R$ and computes the SVD of it $U, S, V^T = SVD(R)$ and then they compute the quantity $UV$ with the comment: ...
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$L_{1,1}$-norm and orthonormal matrices

Given any point $p \in \mathbb{R}^n$ and two matrices whose columns are orthonormal $X \in \mathbb{R}^{n \times j}$, $Y \in \mathbb{R}^{n \times (n - j)}$, such that $Y$ spans the orthogonal ...
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Formulas for Euler Axis components

I am trying to derive the formulas given in this book titled Atmoshperic and Spaceflight Dynamics by Ashish Tewari. I found questions related to finding the rotation matrices online but nothing that ...
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Eigenvectors of a vectorized tranpose matrix

Given the permutation matrix $P\in\mathbb{R}^{n^2}$ such that $P\mathop{\mathrm{vec}}(X)=\mathop{\mathrm{vec}}(X^T)$ for all $X\in \mathbb{R}^{n\times n}$, is there any closed form expression for the ...
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If an orthogonal matrix has determinant -1 then it has -1 as an eigenvalue

I want to show that if a real orthogonal matrix $A$ has determinant $-1$ then $\lambda=-1$ must be an eigenvalue of $A$. I have proven this in a long-winded way and I was wondering if these is a ...
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Number of equations in orthogonal matrix?

I have an orthogonal matrix $A^{n\times n}$, that is $A^TA=I$. What is the number of equations $A$ has to satisfy? It seems to me that the answer is $\frac{n}{2}(n+1)$. My reasoning as follows. Each ...
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Show that $\mathbf{M}^T\mathbf{M}^{-1}\left(\mathbf{M}^{-1}\right)^T\mathbf{M}=I$ where $\mathbf{M}= I - \mathbf{A}$ and $\mathbf{A}^T = - \mathbf{A}$

Let $\textbf{M} = I-\mathbf{A}$ (with $\mathbf{A}^T = -\mathbf{A}$) be a matrix and $\textbf{M}^{-1}$ its inverse. From the identities: \begin{equation} \textbf{M}^{-1}\textbf{M} = I\\ \...
Let $U=\mbox{span}\left\{ \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}\right\}$...