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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1
answer
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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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How to use unitary matrix for single qubit rotations?
I know, that all single qubit unitaries can be written as:
$$
U = \exp\left(-{\rm i}\,\frac{\Theta\,\sigma \cdot n}{2}\right),
$$
where $\Theta$ is an angle, $n$ is a length $3$ direction vector, and $...
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Two sets of vectors having the same Gram matrix
Let $\{ a_k \}$ and $\{ b_k \}$ be two sets of $n$ vectors of the same dimension, such that
$$\langle a_x, a_y \rangle = \langle b_x, b_y \rangle$$
for every $x, y$. Is it true that there exists a ...
- 321
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How to show that this space has same homotopy type as the classifying space of infinite unitary group
To show space $\bigcup_{n\geq 1} \frac{U_{2n}}{U_{n}\times U_{n}}$ has the same homotopy type as $BU = \bigcup_{k\geq 1}BU(k)$, where $BU(k)=\bigcup_{n\geq k} \frac{U_{n}}{U_{k}\times U_{n-k}}$ and $...
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Solution verification: Eigenvalues of unitary matrix
If $A\in M_n(\mathbb{C})$ is unitary, then the eigenvalues of $A$ have module $1$.
$Av=\lambda v\implies \langle Av,Av\rangle=\langle \lambda v,\lambda v\rangle =\lambda\overline{\lambda}\langle v,v\...
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Basis of the decomposition of two commutating groups
$\newcommand{\ket}[1]{|#1\rangle}$
Consider the following representations of the permutation group $S_n$ and the unitary group $U(d)$ acting on the vector space $(\mathbb{C}^d)^{\otimes n}$ like:
$$
P(...
- 23
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Relationship between two matrices whose sum and difference are similar matrices
Let $A$ and $B$ be hermitian $n$-by-$n$ matrices with complex coefficients with the property that $A+B$ and $A-B$ are similar, i.e. there is a unitary $U$ such that $A+B = U(A-B)U^\dagger$. Does it ...
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General lorentz transformation as exponential
I am looking at Lorentz transformations and how the Dirac equation transforms under them. A Lorentz transformation is an element of the matrix group
$$O(1,3) := \left\{\Lambda\in\mathbb{R}^{4\times4} \...
- 514
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Co-block-diagonalization of unitary matrices
I have the following open problem.
For $n\in\mathbb{N}$, consider two complex squared matrices of the same size verifying $A^n=1$ and $A^{\dagger}=A^{n-1}$.
Can you find an integer k such that the two ...
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What does the condition $a^{4} - e^{4 i \varphi} \overline{a}^{4} = 0$ say about a unitary matrix with antidiagonal 0s?
Given a unitary matrix of the form
$$U = \begin{bmatrix}
a & 0 \\
0 & e^{i\varphi}\overline{a}
\end{bmatrix}$$, where $|a|=1$ and $\varphi\in\mathbb{R}$ ($\overline{a}$ is the complex ...
- 179
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isometries and unitary operators, Specht theorem
I was looking at the properties of the trace operator $\operatorname{tr}$, in particular the properties of the trace regarding isometric conjugation.
We say that $T\in \mathcal{H}$ is an isometry if $...
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Trace condition implies matrix is unitary
Question:
Let $U_a$ ($a=1,2,\cdots,n^2$) be unitary $n \times n$ matrices, and suppose that there exists an $n \times n$ matrix $M$ (EDIT: w.l.o.g. we can restrict $M$ to be diagonal[2]) such that
$$\...
- 1,436
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non-isomorphic unitary irreps are orthogonal, and related concepts
Let $ N $ be a normal matrix. Let $ v_i,v_j $ be eigenvectors of $ N $ with eigenvalues $ \lambda_i, \lambda_j $. If $ \lambda_i \neq \lambda_j $ are distinct then $ v_i \cdot v_j =0$ are orthogonal.
...
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