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.
93
questions
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1answer
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Understanding the property of unitary matrix
From wikipedia page https://en.wikipedia.org/wiki/Unitary_matrix Section: Elementary constructions.
What I understand about Unitary matrix is : If we have a square matrix (say 2x2) with complex ...
0
votes
1answer
28 views
Are unitary matrices still unitary under similarity transformations?
$\newcommand\dag\dagger$
I would assume that the property of being unitary is invariant under similarity transformations since similarity transformations are just a change of basis of a linear map, ...
1
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0answers
22 views
Unitary Evolution and Expected Values Clarification (quantum mechanics)
Consider the Pauli Matrices $I = \pmatrix{1\ 0 \\0 \ 1}$, $\sigma_{1} = \pmatrix{0\ 1 \\ 1\ 0}$,
$\sigma_{2} = \pmatrix{0\ -i\\ i\ 0}$,
$\sigma_{3} = \pmatrix{1\ 0\\ 0\ -1}$,
$\bf{Part 1:}$ Compute ...
1
vote
0answers
34 views
Help understanding proof that unitary matrices are length preserving
I'm working through the article Quantum Computing for the Very Curious and am stuck on one aspect of the proof it gives for unitary matrices being length preserving. I've included an annotated ...
0
votes
2answers
74 views
Unitary doesn't carry over isomorphism?
I think I just heard someone say that if $\pi$ and $\rho$ are equivalent representations and if $\pi$ is unitary, then $\rho$ is not necessarily unitary.
Is that really correct?
I would think that ...
1
vote
1answer
42 views
If $UU^*=I$ is the unitary group, what is $(UU^*)^2=1$?
Let $U$ be a $n\times n$ complex matrix satisfying:
$$
UU^*UU^*=I
$$
Does it follow that $UU^*=I$?
Let me have a go at it:
$$
UU^*UU^*=(UU^*)^2=I\implies UU^*=\pm \sqrt{I}
$$
Now the square root ...
0
votes
1answer
24 views
Compactness of unitary group in the symplectic linear group
I am working on the proof of the claim, that
$$U(n) \subset Sp(2n)$$
is the maximal compact subgroup in the book "Introduction to symplectic topology" by McDuff and Salomon.
As far as I see they ...
1
vote
1answer
23 views
Unitary matrix with special relation on matrix entries
Let $n$ be a natural number and let $A = (a^{i}_{j})_{1 \leq i,j \leq n} \in U_{n}$, i.e., $A^{*}A = AA^{*} = I_{n}$. Suppose for each choice of indices $i,j,k,\ell \in \{1,\dots,n\}$ it holds $a^{i}_{...
2
votes
2answers
35 views
Do non unitary involutary matrices exist?
Consider an involutary matrix. This means that the square of the matrix is identity. Identity is clearly a unitary matrix. However, it is not obvious that this matrix should be a unitary one. But, do ...
0
votes
1answer
15 views
If $T$ is normal and given $T$ is invertible, then is $T$ unitary? and Given two unitary operators $U$ and $R$ prove $U^{-1}R$ is also unitary.
Question 1. If $T$ is normal that is $T^{*}T = TT^{*}$, and given $T$ is invertible, then is $T$ unitary?
I am familiar with the result which states the converse. That if T is unitary, T is ...
0
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0answers
30 views
Can we make an algorithm for Schur Decomposition?
Schur's Theorem
If $A$ is any $n \times n$ complex matrix, there exist a unitary matrix $U$ such that
$$U^*AU=T$$
is upper triangular. Moreover, the entries on the main diagonal of $T$ are the ...
0
votes
2answers
47 views
Find unitary matrix given its image.
I'm trying to solve this:
Let $$D_{0}=\begin{bmatrix}
1 & 4\\
2 & 5\\
3 & 6
\end{bmatrix}$$
Find a D such that $$D^{*}D=I$$$$\operatorname{Im}D_{0}=\operatorname{Im}D$$
(D* is complex ...
0
votes
1answer
33 views
Does a Householder Matrix commute with a unitary matrix?
If $A$ is a unitary matrix and $P=I-2\alpha\alpha^H(0\neq\alpha\in\mathbb F^{n\times 1},\lVert\alpha\rVert=1)$, then does $PA$ equals to $AP$?
1
vote
1answer
44 views
$U(n)\cap\mathfrak{u}(n)$ as a submanifold of $\mathbb{C}^{n^2}$
This is a question from an exam on an extra-curricular course in differential geometry.
The task reads "Express $U(n)\cap\mathfrak{u}(n)$ as a submanifold of $\mathbb{C}^{n^2}$, where $U(n)$ and $\...
0
votes
1answer
34 views
If A is unitary and $\det (A^H) = \det(A^*)$ and $\det(A) \det(A^*) = \det(A)^2$ Why can't I say that $\det(A^H) = \pm\det(A)$
this came up on a Homework I have.
I had to prove that the absolute value of the determinant of a Unitary Matrix is 1.
So because $\det (A^H) = \det(A^*)$
and $\det(A) \det(A^*) = \det(A)^2$
as ...
1
vote
1answer
43 views
Is the identity the only matrix unchanged by unitary tranformation?
Say we have a matrix $M$ and we know that $M = UMU^\dagger$ where $U$ is a unitary matrix and $^\dagger$ indicates the conjugate transpose. It is clear that $M = k I $ is a solution, where $I$ is the ...
2
votes
0answers
19 views
Matrices for symmetries of Möbius band
Taken from Groups and Symmetry by M.A. Armstrong, question 18.8:
Setup:
A 5x1 rectangular strip of paper is marked off on both sides into 5 unit squares. The two ends of the paper are then put ...
0
votes
0answers
42 views
Orthonormal basis of eigenvectors of unitary transformation - proof verification
Is my proof of the following proposition correct? The part I would be most unsure about is showing that $A:W^\perp\to W^\perp$, although I don't see any flaws with the argument. A similar proof also ...
1
vote
2answers
60 views
Are the groups $SU(2, ℂ)$ and $U(1, ℍ)$ isomorphic?
Both these groups are double covers of $SO(3, ℝ)$ so they must share the same Lie Algebra at least.
More generally if 2 Lie groups share a Lie algebra, and both are N-covers of another group, are ...
0
votes
0answers
20 views
Spectral norm of non-unitary transformation
Let $X$ be an arbitrary square matrix and $R$ be an invertible matrix of the same size as $X$. $R$ is not unitary. When is it the case that
$$
\| R^{-1} X R \| \le \| X \|,
$$
where $\| \cdot \|$ is ...
2
votes
0answers
50 views
Are the irreducible representations of $U(n)$ the same as $SU(n)$?
I am interested to know whether $U(n)$ has the same (irreducible) representations as $SU(n)$ and why? I have noticed that very often in the literature there is not a clear distinction between these ...
5
votes
1answer
72 views
What are the fixed-point free subgroups of $\mathrm U(2)$?
Let $\mathrm U(2)\subset\Bbb C^{2\times 2}$ be the group of unitary $(2\times 2)$-matrices.
I wonder the following:
Question: What are the maximal fixed-point free subgroups of $\mathrm U(2)$?
A ...
2
votes
2answers
43 views
constructing a unitary representation of $SU(2)$
The group $SU(2)$ acts on $\mathbb{C}^2$. Hence, it has a representation on the space of all functions on $\mathbb{C}^2$. In particular, let $V_k$ be the space of homogeneous polynomials of degree $k$....
1
vote
1answer
49 views
Matrix exponentiation of a Kronecker product of Pauli matrices
I need to numerically compute the matrix-exponential of a Kronecker product of Pauli matrices (including the identity).
For example,
$$
\exp( X \otimes Y \otimes I \otimes Z \;\otimes \;... )
$$
or ...
1
vote
1answer
42 views
Unitarily invariant norm of matrix
Let $A \in \mathbb{C}^{n \times n}$, $X \in \mathbb{C}^{n\times k}$, $M\in \mathbb{C}^{k\times k}$. If $X^HX=I_k$ and $\|\cdot\|$ is unitarily invariant, i.e., if for any unitary matrices $U, V$ we ...
1
vote
0answers
18 views
How to generate random sparse unitary with given density?
For my research, I need to generate sparse (complex-values) unitary matrices at random from a uniform distribution. It is not a problem for me to generate the generic unitary matrices using, e.g., ...
1
vote
1answer
19 views
A matrix $g$ is in the Unitary Group iff $g \bar g^t=I$
The Unitary group was defined as follows:
$U_n = \{g \in GL_n(\mathbb{C}):<gu,gv>=<u,v> u,v\in \mathbb C^n\}$
Where $<u,v>=\sum_1^n u_i \bar v_i$
I'm trying to prove that a matrix $...
2
votes
0answers
31 views
What is U(1)/Z2 isomorphic to?
Probably a very easy question but I can't seem to figure it out: What is the group $\frac{U(1)}{\mathbb{Z_2}}$ isomorphic to?
My intuition is telling me to use the homomorphism theorem, so I want to ...
4
votes
2answers
78 views
Abelian subgroup of the unit quaternions
Let $\Bbb S^0(\Bbb H)\cong \mathrm{SU}(2)$ denote the multiplicateive group of unit quaternions.
This group is non-abelian.
Of course, the subgroups generated by as $\def\<{\langle}\def\>{\...
0
votes
0answers
28 views
If $A$ and $B$ are unitarily equivalent, then $\sum_{i,j}|A_{ij}|^2 = \sum_{i,j}|B_{ij}|^2$
If $A$ and $B$ are unitarily equivalent, then $\sum_{i,j}|A_{ij}|^2 = \sum_{i,j}|B_{ij}|^2$
We have a unitary matrix $U$ such that $U^*AU = B$
How can I proceed from here to show $\sum_{i,j}|A_{ij}|^...
3
votes
1answer
62 views
Quaternions and $SU(2)$
I want to show that the following map is in $SO(\mathbb{H})$.
Let $A,B\in SU(2)$ define $F:H\rightarrow H$ by $F(h)=AhB^{-1}$.
Intuitively this makes sense since $SU(2)$ can be identified with the ...
1
vote
0answers
47 views
Fundamental group of $U(n) / \mathbb{Z}_m$
I would like to know the fundamental group of $U(n) / \mathbb{Z}_m$, where the $\mathbb{Z}_m$ in the quotient is the diagonal matrices with entries equal to an $m$th root of unity.
The strategy I've ...
1
vote
1answer
56 views
Trace inequality of the form $\operatorname{Tr}\Big(QMQ^*f(M)\Big) \le \operatorname{Tr}\Big(QQ^*Mf(M)\Big)$
Let $Q$ be unitary and $M$ be Hermitian positive semidefinite (psd), so that $QMQ^*$ is also Hermitian psd. What conditions (probably on $f(M)$)
makes the trace inequality $$\operatorname{Tr}\Big(QMQ^*...
4
votes
2answers
78 views
Unitary Operators on
Let $L^2_{\mathbb{P}}(\Omega)$ be a separable Hilbert space and $(\Omega,\Sigma,\mathbb{P})$ be a probability space. Given two $f,g \in L^2_{\mathbb{P}}(\Omega)$ is there a unitary (or self-adjoint) ...
1
vote
0answers
29 views
Classification of closed subgroups of U(2)
Do we know a complete classification of all closed subgroups of $U(2)$.
1) upto isomorphism
2) upto conjugation
If so could someone provide me a reference as to where I might find this or some ...
1
vote
1answer
102 views
Proof that any unitary matrix can be represented as $e^{i\theta_1 Z}e^{i\theta_2 X}e^{i\theta_3 Z}$
Given the Pauli matrices $X$ and $Z$ defined as
$$X = \begin{pmatrix}0 & 1\\ 1& 0 \end{pmatrix},\ \ Z = \begin{pmatrix}1 & 0\\ 0& -1 \end{pmatrix}$$
I read a result (see page 2) ...
0
votes
1answer
61 views
Unitary matrices are invertible
By definition, an unitary matrix $U$ is defined as $U U^H = I$.
How can we prove that an unitary matrix $U$ is invertible? It can happen that $U^H U \neq I$.
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0answers
21 views
What is the difference between compositional inverse and multiplicative inverse in Quantum mechanic?
Really am mixed , I have read a definition of bounded linear operator as it defined below
A bounded linear operator $U: H \to H$ on a Hilbert space $H$ is called a unitary operator if it satisfies $...
1
vote
0answers
22 views
Local unitary operators preserving a specific subspace should be permutation matrices.
I have the following question coming from quantum information theory. Consider linear space over $C$ of dimension 3, and a tensor product of three such spaces. So, there are 27 basis vectors of the ...
2
votes
1answer
39 views
Homotopy in $U_{2}$
Let $\mathbb{S}^{1}$ be the unit circle and $U_{2}\subset M_{2\times 2}(\mathbb{C})$ be the unitary group. Let $f,g:\mathbb{S}^{1}\rightarrow U_{2}$ be maps defined by
\begin{align}
f(x)=
\begin{...
0
votes
1answer
22 views
Inverse of unitary linear transfomation
A linear transformation that maps $|0\rangle$ to $ \alpha |0\rangle+ \beta |1\rangle$ and $|1\rangle$ to $\gamma|0\rangle +\delta |1\rangle$ is unitary if $|\alpha|^2+|\beta|^2=1$ and $|\gamma|^2+|\...
3
votes
1answer
45 views
Does every bistochastic matrix $A$ have a unitary matrix $U$ s.t. $a_{ij} = |u_{ij}|^2$?
Bistochastic matrix is a square matrix made of non-negative reals s.t. the sum of elements of any row or column equal 1.
Unitary matrix is a square matrix with complex entries s.t. $UU^*=E$
It's ...
3
votes
4answers
64 views
Find the reflection of the point $(4,-13)$ in the line $5x+y+6=0$
Find The image(or reflection) of the point $(4,-13)$ in the line $5x+y+6=0$
Method 1
$$
y+13=\frac{1}{5}(x-4)\implies x-5y-69=0\quad\&\quad 5x+y+6=0\implies (3/2,-27/2)\\
(3/2,-27/2)=(\frac{x+4}{...
1
vote
1answer
34 views
When is there an inner product making a given matrix unitary?
The unitarity of a given operator/matrix $A$ depends on the underlying inner product, so I guess it in principle possible for a given non-unitary $A$ to act as a unitary with respect to a different ...
2
votes
1answer
141 views
Hermitian (?) representations of $su(2)$
I'm searching for the irreducible representations of $su(2)$. To be clear on my setting :
$su(2)$ is the real vector space of skew-hermitian traceless matrices.
I'm searching for the irreducible ...
0
votes
1answer
44 views
Is $SU(n) \subset \text{Spin}(2n)$?
I believe this should be true for the following reason:
The injection $SU(n) \hookrightarrow SO(2n)$ induces a map of Lie algebras $\mathfrak{su}(n) \hookrightarrow \mathfrak{so}(2n) \cong \mathfrak{...
4
votes
2answers
306 views
Prove this matrix to be unitary
This is a homework question so, hints are appreciated. But if someone is generous enough, to show the full calculation, I'd be quite grateful!
Say a matrix B is anti-hermitian:$$\begin{bmatrix}
i &...
1
vote
1answer
85 views
Verify the matrix exponential $e^{i\hat{H}t/\hbar}$ is unitary
What could be an example of an anti-hermitian matrix $i\hat{H}t$ , which satisfies the matrix exponential $$e^{i\hat{H}t/\hbar}$$ being a unitary matrix?
-3
votes
1answer
45 views
Proof that $\dfrac{\partial U_t}{\partial t} U_t^\dagger$ is anti-Hermitian by integration by parts [closed]
A proof to understand why $$\frac{\partial U_t}{\partial t} U_t^\dagger$$ is anti-Hermitian.
0
votes
2answers
45 views
why is this vector multiplication move valid?
This is dealing with properties of unitary matrices and this is proving that
eigenvectors corresponding to different eigenvalues are orthonormal.
start with $Ux = \lambda_1 x$ and $Uy = \...
