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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&...
Andrius Kulikauskas's user avatar
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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....
Andrius Kulikauskas's user avatar
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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 ...
Cole's user avatar
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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 $...
Curious's user avatar
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2 answers
50 views

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 ...
NYG's user avatar
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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 $...
Rkb's user avatar
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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\...
user926356's user avatar
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2 votes
1 answer
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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(...
CaLa's user avatar
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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 ...
extempore's user avatar
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1 answer
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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} \...
Rasmus's user avatar
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1 answer
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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 ...
Hunfail Karta Hunfail505's user avatar
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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 ...
upe's user avatar
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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 $...
ana's user avatar
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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 $$\...
Ruben Verresen's user avatar
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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. ...
Ian Gershon Teixeira's user avatar
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Issue with proof that SU(2) conjugation on Pauli matrices acts via SO(3)

I want to prove that for a given $U\in SU(2)$, $\phi_U(\vec x)=U\vec x \cdot \vec \sigma U^{\dagger}$ can be identified with a $R(U)=R\in SO(3)$ such that $\phi_U(\vec x)=(R\vec x)\cdot \sigma$. ...
user62783's user avatar
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Matrix representation of Möbius transformation from unit disc to upper half plane.

The task is as follows: Verify that the Möbius transformation $ z \mapsto \frac{iz + i}{-z + 1} $ from the disc model of the hyperbolic plane to the upper half-plane model may be defined by an ...
vencint's user avatar
1 vote
1 answer
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Why ins't this SU(4) Matrix produced by the exponential map?

I was working with the SU(4) Lie group, which is compact and simply connected. This should imply that the exponential map is sujective on the group. However i came across the matrix $$G=\begin{pmatrix}...
Arthur's user avatar
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2 answers
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How should we characterize the relationship between two matrix representations of a linear operator with respect to two different orthonormal bases?

Nielsen / Chuang remark on page 71 of "Quantum Computation and Quantum Information" that, if $| v_i \rangle$ and $|w_i \rangle$ are orthonormal bases, then the operator $U$ defined by $\sum_{...
mchk's user avatar
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If $U $ is a unitary linear operator, how can I show that any matrix representation of $U$ must be a unitary matrix?

Nielsen / Chuang "Quantum Computation and Quantum Information" states on p. 70: "A matrix $ U$ is said to be unitary if $U^\dagger U = I$. Similarly, an operator $U$ is unitary if $U^\...
mchk's user avatar
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1 answer
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Commutators of unitary matrices and their unitary square roots with other matrices

Given some unitary matrix $U$, we can possibly find many unitary square root matrices $S_i$ so that $S_i^2=U$. Let's assume we have an additional (complex) matrix $T$ so that it commutes with $U$: $$ [...
sqrt6's user avatar
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Homogeneous space of the unitary group U(n)

Consider the unitary group $U(2^n)$ and let $G$ be the subgroup isomorphic to $U(2^{n-1})$ embedded as: \begin{pmatrix} U & 0\\ 0 & U \end{pmatrix} where $U \in U(2^{n-1})$ (the same matrix in ...
mkk's user avatar
  • 388
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1 answer
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Is this basis for complex matrices necessarily a unitary basis?

Let us consider the vector space of complex $n \times n$ matrices. Let $\{ V_i \}_{i=1,2,\cdots,n^2}$ be a trace-orthogonal basis of matrices, i.e., $$ \forall i,j \in \{1,2,\cdots,n^2\} : \quad \...
Ruben Verresen's user avatar
1 vote
1 answer
52 views

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 ...
Rell's user avatar
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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 ...
Damuna Taliffato's user avatar
-4 votes
1 answer
159 views

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 ...
herashefat's user avatar
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1 answer
62 views

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 ...
Squirrel-Power's user avatar
6 votes
2 answers
186 views

Calculating the Haar integral on $SU(2)$ in practice

I'm trying to calculate the Haar integral on $SU(2)$ of a given function $f: SU(2) \to \mathbb{R}$. For this particular function, I know the value of it on the subgroup $$T = \left\{ \begin{pmatrix} z ...
Robin's user avatar
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1 answer
95 views

How do i prove that this matrix is unitary?

I'm having trouble with an algebra exercise. I'm supposed to determine whether the matrix $U$ given by: $$ U = \frac{1}{\sqrt{n}} \pmatrix{u_{0,0} & u_{0,1} & ... & u_{0,(n-1)}\\ u_{1,0} &...
Luca Seggiani's user avatar
1 vote
1 answer
33 views

How to obtain the following equality: $EE' = (-1)^{a.b' + a'.b}E'E$

I am having difficulty understanding why: $$EE' = (-1)^{a.b' + a'.b}E'E$$ $E$ and $E'$ are error operators of the form: $$E = i^{\lambda} X(a)Z(b)$$ $$E'=i^{\lambda '} X(a')Z(b')$$ where $\lambda \in ...
am567's user avatar
  • 329
1 vote
2 answers
223 views

General form of a $4\times 4$ unitary matrix

I was trying to find the most general way of writing a $4\times 4$ unitary matrix but I got stuck when I obtained a set of equations that did not look very pretty after starting from something like: $$...
Amentia's user avatar
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2 votes
3 answers
299 views

Weights in SU(3)

I'm a physicist trying to understand the concept of weight vectors in SU(3). In my professor's notes, weights are defined as the eigenvalues of the elements of the basis, which can be simultaneously ...
Gorga's user avatar
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1 vote
1 answer
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Condition on unitary operator for real eigenstates of Hamiltonian

I'm working with the discrete-time quantum walk in which the evolution is described by the unitary operator - $$U = S(I\otimes C)$$ where $C$ is the coin operator (acts on spin degree of freedom of ...
Young Kindaichi's user avatar
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0 answers
25 views

Maximum and minimum value of trace of unistochastic matrix

Let $P$ be a unistochastic matrix, that is, there exists a unitary matrix $U$ such that $P_{ij}=|U_{ij}|^2$. What is the maximum and minimum value of $\mathrm{Tr}[P]$? I'm also interested in the ...
ytaguchi's user avatar
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0 answers
48 views

Size of an $\epsilon$-net on n-qubit unitaries

I am currently reading through Computational Quantum Entanglement paper and there is a following statement there in proof of Lemma 4.1 We then use that an $\eta$-net (this is an $\epsilon$-net with ...
Piotr Lewandowski's user avatar
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Power series of unitary matrices that is also unitary

Given a unitary matrix $U$ such that $U^{\dagger} U = UU^{\dagger} = I$, I would like to examine a finite power series $V = \sum\limits_{i=0}^n\alpha_i U^{i}$ where $\alpha_i \neq 0$. Can this ...
user1936752's user avatar
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$F_{s;I}=\frac1{2^{n-|I|}}\sum_{supp(t)\subset I^c}tr(\chi_tF_{s;I})\chi_t=\frac1{2^n}\sum_{supp(t)\subset I^c}tr(\chi_s\otimes\chi_tf)\chi_t$

Let $f$ be a unitary operator. $\chi_s$ is the stabilizer operator where $s\in \{0,1,2,3\}^n$ where $\chi_s=\bigotimes\limits_{i=1}^n\sigma^{s_i}$ where $s_i\in \{0,1,2,3\}$. So we can write $f=\sum\...
Soham Chatterjee's user avatar
0 votes
1 answer
61 views

Are irreducible representations of the (complex) projective unitary group irreducible representations of the unitary group?

It is convenient to label irreps of the unitary group by highest-weight vectors $\lambda = (\lambda_1, \lambda_2,...\lambda_n)$ satisfying $\lambda_1 \geq \lambda_2\geq...\lambda_n)$. Then we obtain a ...
ors's user avatar
  • 342
0 votes
1 answer
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On the set of complex square matrices $H$ such that $H + H^* = I$

I am interested in the set of $n \times n$ complex square matrices $H$ such that $H + H^* = I$. Here, $H^*$ is the conjugate transpose of $H$ and $I$ is the $n \times n$ identity matrix. These ...
trillianhaze's user avatar
1 vote
0 answers
136 views

Geodesic distance in the (complex) projective unitary group

I think this information should be well known, and in a textbook somewhere but I have been unable to find it so I apologise if this is standard material. I am a physicist and trying to do some ...
ors's user avatar
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0 answers
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Is this operator unitary?

It is well know that the operator $\exp\left[-i\Delta t\hat{H}\right]$ is a unitary operator if $\hat{H}$ is an Hermitian operator. The proof is based on Taylor expanding and affecting the dagger: $\...
Yair's user avatar
  • 501
0 votes
1 answer
99 views

What is the unitary subset of Lie algebra $su(2)$?

Lie algebra $su(2)$ consists of the $2\times 2$ skew-hermitian complex matrices with addition and multiplication by real numbers as vector space operations and commutator as Lie bracket. The i-...
mma's user avatar
  • 2,065
2 votes
1 answer
52 views

Tensor product and unitarity

If we consider a $4\times 4$ complex unitary matrix $V\otimes V\in U(4, \mathbb{C})$, and $V\in M_{2\times 2}(\mathbb{C})$, does this imply that $V$ is also unitary? I am aware that $V\otimes V\in U(4,...
user avatar
1 vote
1 answer
70 views

Show that $U^* A U$ is upper triangular, with $A$ upper triangular, and $U$ unitary and lower-Hessenberg

Let $A \in \mathbb{C}^{n \times n}$ be upper triangular. Let $u$ be a unit-norm eigenvector of $A$ whose eigenvalue is $a_{11}$ (the top-left entry of $A$). Let $U \in Unitary(n)$ be a lower-...
ccriscitiello's user avatar
1 vote
1 answer
51 views

Unitary transformation to triangularize 2 matrices at the same time

I am trying to show that: Given 2 complex non-singular matrices $A$ and $B$, there exist two unitary matrices $Q$ and $Z$ s.t. $Q^*AZ$ and $Q^*BZ$ are simultaneously upper triangular. Thank for your ...
Gianmichele Palumbo's user avatar
5 votes
1 answer
74 views

Unitary operators and a product of reflections

The book Advanced Linear Algebra by Steven Roman states that Reflections or Housholder transformations are self-adjoint and unitary. Moreover, Theorem 10.17 of this book states every unitary $\tau \in ...
khashayar's user avatar
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1 vote
0 answers
63 views

Finitely generated dense subgroup of an unitary group

I am interested in finitely generating dense subgroups of the unitary group $U(d)$ of order $d$ (with respect to the operator norm topology), and am considering the following question. Let $I_d$ be ...
trillianhaze's user avatar
2 votes
0 answers
42 views

Triple Product Formula on $K = SU(2)$

Let $K = SU(2) = \{ k[\alpha ,\beta] | \alpha ,\beta \in \mathbb{C}, |\alpha |^2 + |\beta |^2 = 1 \} $ with $$ k [ \alpha , \beta ] = \begin{pmatrix} \alpha & \beta \\ - \overline{\beta} & \...
Misaka 16559's user avatar
1 vote
0 answers
65 views

First fundamental theorem of invariants for the unitary group

Setting Let $V$ transform according to the direct representation of the unitary group $U(d)$. I have a polynomial on $P:V^k \times (V^\star)^l\rightarrow \mathbb R$ where $V^\star$ is the conjugate ...
Tom's user avatar
  • 11
0 votes
1 answer
124 views

Lie algebra of real unitary matrices

The Lie algebra associated to the group $SO(n)$ of real-valued special orthogonal matrices, is given by the set $\mathfrak{so}(n)$ of anti-symmetric real-valued matrices equipped with the commutator. ...
Meths's user avatar
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