# 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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### Is spectral theorem an equivalence?

A matrix $A \in K^{n \times n}$ is diagonalizable with orthonormal basis on $K$ if there exists $P \in K^{n \times n}$ orthogonal such that $P^*AP = D$ where $D \in K^{n \times n}$ is a diagonal ...
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### Invertibility of Similar and Orthogonal Matrices [closed]

Are matrices similar to orthogonal matrices invertible?
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### Finding the spectral decomposition of a given $3\times 3$ matrix

I am trying to find the spectral decomposition of the following matrix: $$A=\begin{pmatrix} 4 & 0 & 0\\ 0 & 2 & 0\\ 2 & 3 & 0\\ \end{pmatrix}$$ So I found the eigenvalues ...
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### Show proofs for inverse of these singular matrix.

I tried really hard, but I have no idea how to approach this question. A and B matrix are not invertible, so inverse does not exist. So, how do I go about proving them ? Simply saying they do not have ...
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### Let $X, Y ∈ R^{n × m}$. We are looking for an orthogonal matrix $R ∈ M_n (R)$ that minimizes $\|RX - Y\|^2=\operatorname{tr}((RX - Y)^T (RX - Y))$.

I'm trying to show that the wanted matrix is $R = VU^T$, where $XY^T = UΣV^T$ is the singular value decomposition, but I'm having trouble understanding what it means or how to start.
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### Show that $O_2(\mathbb{R})$ contains only rotational and reflective symmetries.

Show that $O_2(\mathbb{R})$ contains only rotational and reflective matrices. I know that rotational and reflective symmetries are part of $O_2(\mathbb{R})$. I want to show that there is no other ...
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### Definition of orthogonal projection of a matrix

Given an $m$ x $k$ orthogonal matrix $Q$ and an $m$ x $m$ matrix A, I know that the matrix $QQ^{T}A$ is the orthogonal projection of $A$ onto the column space of $Q$. Recently, in a paper, I read that ...
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### $SU(2)$ and the $3$-sphere, with $SO(2)$ a longitude

I'm currently studying the connection between $SU(2)$ and the $3$-sphere. A longitude on the sphere is of the form $QTQ^*$ where $Q \in SU(2)$ and $T = \{D_\lambda : \lambda \bar{\lambda} = 1 \}$, ...
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### Is an orthogonal matrix with determinant $-1$ a rotation matrix? [closed]

I know that an orthogonal matrix with determinant $1$ is a rotation matrix. However, is an orthogonal matrix with determinant $-1$ also a rotation matrix?