# Questions tagged [unitary-matrices]

This tag is for questions relating to Unitary Matrices which are comprise a class of matrices that have the remarkable properties that as transformations they preserve length, and preserve the angle between vectors.

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### Explicit example of a set of coset representatives of $U(n)$ within $O(2n)$

I understand how to identify a unitary group $U(n)$ with the elements of the orthogonal group $O(2n)$ which commute with a linear complex structure $J$. I am also aware of the "two-out-of-three&...
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### Conjugacy action of $SO(2m)$ on $O(2m)/U(m)$

I seek intuition about the symmetric space $S$, the set of orthogonal complex structures in $\mathbb{R}^n$ for even $n=2m$. I am finding J.H.Eschenburg's Lecture Notes on Symmetric Spaces very helpful....
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### On equivalence of unitary matrices and unitary operators

I'm self-studying Axler's LADR and am working on the section on Unitary Operators. He defines a unitary operator as an invertible isometry. We've proved the following equivalences: He then defines a ...
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### Inequalities on the trace of matrix products

For two $n\times n$ hermitian matrices $A$, $B$, we have the trace inequality $$\text{tr}(AB)\leq\sum_{i=1}^{n}\lambda_i(A)\lambda_i(B)$$ where the $\lambda_i(X)$ are the eigenvalues of X ordered in ...
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### Generation of Hermitian invertible matrix with fixed number of non-zero elements

I have a Hermitian matrix that is invertible, that is I can write it as: $H = U^\dagger D U$, where $U$ is a unitary matrix, and $D$ is a diagonal matrix with the eigenvalues of $H$, which must be ...
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### To find unknown rows in a unitary matrix

The problem is to find a unitary matrix A whose first row is a multiple of a) $(1,1,-i)$ and b) $\left(\frac{1}{2},\frac{i}{2},\frac{(1-i)}{2}\right)$ Now the first part of a is easy because the rows ...
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### Show that the unitary group $U(n)$ and the special unitary group $SU(n) \times S^1$ are not isomorphic as Lie groups when $n > 1$ [duplicate]

Show that the unitary group $U(n)$ and the special unitary group $SU(n) \times S^1$ are diffeomorphic as manifolds, but not isomorphic as Lie groups when $n > 1$. The above question is from a ...
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