# 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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### If $A$ is an orthogonal real matrix that commutes with all orthogonal matrices then $A$ is a scalar multiple of identity matrix.

If $A$ is an orthogonal real matrix that commutes with all orthogonal matrices then $A$ is a scalar multiple of identity matrix. I was able to prove the previous part of the question which is the ...
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### Invertibility proof of (I+A) [on hold]

If p(M)< 1 such that (I - M) is invertible, can the same be said for (I+M)?
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### Infinitesimal increments to independent variables in orthogonal matrices

I'm trying to read the proof for existence of the Singular Value Decomposition from the Eckart-Young (1936) paper. In page 215, the authors mention "if $u$ is any orthogonal matrix and the ...
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### Alternative way to show that the special orthogonal group is compact

To show that the special orthogonal group ${\rm SO}(n,\mathbb R)$, carrying the subspace topology induced by ${\rm Mat}_{n}(\mathbb R) \cong {\mathbb R}^{n}$, is compact many proofs use the Heine-...
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### Householder matrices for thin QR updating

Suppose you have a QR decomposition of the form: $X=QR$ (Where $X$ is an arbitrary matrix of size $(n \times p)$, $Q$ is an orthogonal matrix of size $(n \times n)$ and $R$ is an upper trapezoidal ...
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### 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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### Orthogonal projection matrices to the subspaces ${Ker(X)}$ and ${Ker(X)^{\perp}}$ [closed]

How to determine the orthogonal projection matrices to the subspaces ${Ker(X)}$ and ${Ker(X)^{\perp}}$ , if $Ker(X)=span(v)$, where it is $v≠0$ ?
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### Dimension of the orthogonal algebra?

The following is on page 3 of Introduction to Lie Algebras and Representation Theory by Humphreys: Here the author claims that the dimension of the orthogonal algebra is $2l^2+l$; but I think the ...
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### Find a unitary basis of the $\mathbb{R}$-vector space of $n \times n$ (complex) Hermitian matrices

The question is on the title ($n \in \mathbb{N}^*$). To be clear, unitary basis here means basis consisting of (complex) unitary matrices. I wonder this question because recently I've read about ...