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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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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}_{...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $AP$?
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$U(n)\cap\mathfrak{u}(n)$ 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 $\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $k$....
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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 ...
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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 ...
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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., ...
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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 $...
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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 ...
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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\>{\...
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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}|^...
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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 ...
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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 ...
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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^*...
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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) ...
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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 ...
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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) ...
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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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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 $...
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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 ...
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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{...
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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+|\...
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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 ...
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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}{...
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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 ...
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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 ...
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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{...
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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 &...
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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?
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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.
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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 = \...