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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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-...
- 607
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Unitarily similar of 2*2 matrices [closed]
Are the matrices $A=\begin{bmatrix} a&b\\0&c\end{bmatrix}$ and $B=\begin{bmatrix} c&b\\0&a\end{bmatrix}$ unitarily similar? In fact, can we find a unitary matrix $U\in M_2(\mathbb{C})$ ...
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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 ...
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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 ...
- 2,072
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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 ...
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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} & \...
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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 ...
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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. ...
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question about unitary matrix and PSD matrix
B,C are two tall matrices but are not assumed to have full column rank. Show that $BB^H=CC^H$ if and only if $C=BQ$ for some unitary matrix Q with k x k dimension.
I try to use EVD on both $CC^H$ and $...
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Difference Between Simultaneous Diagonalization and Generalized Eigenvalue Problem
I am wondering about the difference between simultaneous diagonalization and the general eigenvalue problem. Here is my understanding of simultaneous diagonalization and the general eigenvalue problem:...
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Spectral radius of $\sum_{i=1}^N U_i \otimes J_i \otimes U^*_i \otimes J^*_i$ bounded by that of $\sum_{i=1}^N J_i \otimes J^*_i$ for unitary $U$
Consider a set of $n$ by $n$ unitary matrices $U_i$ and a set of arbitrary $m$ by $m$ matrices $J_i$, with $i=1,...,N$.
Consider the eigenvalues $\{ \lambda_{U \otimes J} \}$ of the operator $O_{U \...
- 1,522
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Cauchy-Schwarz-like inequality for certain determinants
Let $U = [u_1 | \ldots | u_n]$ be a special unitary matrix (meaning that $u_i$ is its $i$-th column) and let $A$ be an $n$-by-$n$ hermitian matrix, where $n \geq 3$. I am interested in the following &...
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Is it possible to find unitary $\boldsymbol{\Theta}$ and scalar $\lambda$ such that $\boldsymbol{\Theta a} = \lambda \boldsymbol{b}$?
Given vectors $\boldsymbol{a}$ and $\boldsymbol{b} \ne \boldsymbol{0}$ in the same Hilbert space, does
$$\boldsymbol{\Theta a} = \lambda \boldsymbol{b}$$
always hold for some unitary $\boldsymbol{\...
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circulant Schur decomposition
Let $A_{1},A_{2},\dots,A_{m}$ be arbitrary $n\times n$ complex matrices.
Prove that there are $n\times n$ unitary matrices $Q_{1},Q_{2},\dots,Q_{m}$ such that matrices $$Q_{1}^{*}A_{1}Q_{2},\ Q_{2}^{*...
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Unitary operator statement
I encountered this statement in the book
$$W(U^{*}TU)=W(T)$$ for any unitary U where $W(T)$ is the numerical range. Definition of the numerical range is $$W(T)=\{\langle Tx,x\rangle |x\in H,\|x\|=1\}$$...
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Need a book about complex hermitian matrix and special unitary matrices and their optimization in numerical problems
I am studying about tomographic interferometric sar polarimetry and i don't have enough knowledge about complex hermitian matrix , special unitary groups, their quadratic forms and numerical ...
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Eigenvalues of $A \overline{A}$
Let $A \in \mathbb{C}^{n \times n}$, with $A_{ij} = a_{ij} \in \mathbb{C}$. I am mostly interested in the case $n =3$, but a general pattern is fine too. Define
$$(\overline{A})_{ij} := \overline{a_{...
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Matrix multiplication involving a unitary matrix in exponential form and a simple matrix
I am not sure if this is even a valid question, but it will help to clear my doubts anyway.
For context, check out Violating the Thermodynamic Uncertainty Relation in the Three-Level Maser.
So I am ...
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Under what conditions does $B = A B A^\intercal$ imply $A = I$ for a symmetric matrix B [duplicate]
I'm not sure this is generically true, but as part of something I'd like to prove I'm hoping to make use of something along the lines of being able to write for some symmetric, nonzero $B$
$$
B = A B ...
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The dimension of the unitary matrix as a real variety or a complex variety.
For a complex matrix $O\in \mathbb{C}^{n\times n}$ which satisfies $O^*O=I$, it can be viewed as a real affine variety in $ \mathbb{R}^{2 n^2}$. Let $U$ be the real part of $O$ and $V$ be the image ...
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Modified orthogonal relation between unitary matrices with columns multiplied by a complex factor
Suppose $U$ is a unitary matrix of dimensions $n\times n$; then we know ($U^{\dagger}$ is conjugate transpose):
\begin{eqnarray}
\sum_k U_{nk}U_{km}^{\dagger}=\delta_{nm}.
\end{eqnarray}
How does this ...
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SVD of product of diagonal and unitary matrices
Given two (possibly rectangular) diagonal matrices $\Sigma_\text{L}$ and $\Sigma_\text{R}$ with nonnegative elements, what can we say about the singular value decomposition of
$$\Sigma_\text{L} X \...
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su(2) vs ZXZ decomposition
I know that, if we consider
$$e^{i \alpha Z} e^{i \beta X} e^{i \gamma Z}$$
(where $X, Z$ are the Pauli matrices) then we can get any element of $SU(2)$ (the so-called $ZXZ$ decomposition). If I write ...
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Existence of unitary operator affecting partial trace? [closed]
Let $\rho$ and $\phi$ be any two different density matrices on $H_A \otimes H_B$ such that $Tr_B (\rho) = Tr_B (\phi)$.
Does there always exist a unitary $U$ on $H_A \otimes H_B$, $U\rho U^{\dagger} = ...
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A formula for a unitary matrix
Proposition: U = (H+iI)(H-iI)^-1 is unitary if H is self-adjoint. I'm having trouble finding a proof. It's straightforward if (H-iI) and (H+iI)^-1 commute. But I don't see it.
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Exercise 4.9 Nielsen and Chuang's "Quantum Computation and Quantum Information
I am trying to work through an online solution to Exercise 4.9 in Nielsen and Chuang's "Quantum Computation and Quantum Information:
The question is:
Explain why any single qubit unitary operator ...
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Prove that set of matrices is dense in $U(2)$
Consider the group of matrices $B$ generated by taking products of the matrices
\begin{equation}
\rho_1 = \begin{pmatrix}\exp(-4\pi i/5) & 0\\ 0 & \exp(3\pi i/5)\end{pmatrix}\\
\rho_2 = \begin{...
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Implication of mean ergodic theorem
My doubt is the following. Mean ergodic theorem tells us that (there are several versions of the theorem, I will write it down as involving unitary operators on a Hilbert space just because it is ...
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What is the geometric meaning of the condition $det(A) = 1$ for $A \in SU(n)$?
Let $A$ be a real square matrix. The condition that $A$ preserves the inner product, and hence preserves lengths, is the interpretation of $A \in O(n)$. If we further require that $det(A) = 1$, then ...
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Generators of $SU(n)$
The generators of $SU(N)$ have the following properties:
$Trace[T^{a}T^{b}]=\frac{1}{2} \delta^{ab}---(i)$ and $Trace[T^{a}]=0---(ii)$.
In product form $$T^{a}T^{b}=\frac{1}{2}(\frac{1}{N} \delta^{ab} ...
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Unitary matrix of polar decomposition
I know unitary matrix of polar decomposition ($A=UP$; $U$ is unitary and $P$ is positive semi-definite) cannot be unique (but $P$ is!) if $A$ is not invertible.
Can you give some examples that $A=UP=...
Semin Kwak
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Question on unitary matrices
If two complex numbers $a, b$ are such that $|a| = |b|$ then there is a unitary complex number $e^{i\theta}$ such that
$$a = b e^{i\theta}$$
Does this hold also for matrices? My best guess is that we ...
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Given $A \in \Bbb C^{3 \times 3}$ that has a SVD $A=u_1v_1^*+u_2v_2^*+u_3v_3^*$ then is $A$ a unitary matrix?
Given $A \in \Bbb C^{3 \times 3}$ that has a SVD $A=u_1v_1^*+u_2v_2^*+u_3v_3^*$ then is $A$ a unitary matrix?
We know that every matrix $A \in \Bbb C^{m \times n}$ has a SVD. Some things that I am ...
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If $A$ is Hermitian and $U=(A-iI)(A+iI)^{−1}$ is unitary, then $U-I$ is invertible
I'm stuck on the $3$rd part of a question, the first part was proving $A-iI$ is invertible when $A$ is Hermitian, second was proving $U=(A-iI)(A+iI)^{−1}$ is unitary. Now I'm being asked to prove $U-I$...
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Diagonalize a Unitary matrix in Fourier domain [closed]
Given a unitary matrix U which I know there exists a unitary diagonal matrix D (unit circle elements in diagonal) such that DFT2(UD) is diagonal (i.e there exists another unitary diagonal matrix V ...
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QR Factorization of Power of Matrix
If we have the QR decomposition of $A^p = Q_p R_p$, can we say anything about the matrix $Q_p^T A Q_p$?
For $p=1$, we clearly have that $Q^T A Q = RQ$. However, even at $Q_2^T A Q_2$, I don't really ...
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Trace of a multiplication operator
I am studying an operator $A$ that acts as a multiplication by a matrix $A(k)$ and am trying to find a relation between the trace of the operator $A$ and the trace of the matrix $A(k)$. However, ...
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Unitary group and subgroup are non-isomorphic
Consider the group $U(n)$ of unitary $n\times n$ matrices, as well as its subgroup $V$ of $n\times n$ matrices which are obtainable by permuting rows of diagonal matrices $\mathrm{diag}(a_1,\dots,a_n)$...
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Real matrix as sum of unitary matrices
I want to write a real matrix $M$ as a sum of unitary (or symmetric) matrices.
$M$ can be broken down into a diagonal and skew-symmetric matrix:
$$M = D + A$$
$$D = \begin{bmatrix} D_1 & 0 & 0 ...
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Family of $n!+1$ non-commuting matrices in unitary group
I am looking to find $n!+1$ matrices in the unitary group $U(n)$ which do not pairwise commute. I believe that the $n!$ matrices given by starting with the $n\times n$ matrix $$U:=\begin{pmatrix}-1&...
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How to compress a unitary matrix?
I have a set of unitary matrices (four-dimensional, but would like to get an answer in the general case). I get it from singular value decomposition (svd).
There are a lot of matrices, and I began to ...
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0
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Parametrizing Unitary matrices using Hermitian matrices. Covers the whole space?
An $n\times n$ unitary matrix can always be written in the form,
$$ U=e^{i\,H}\,,$$
where $H$ is a Hermitian matrix. If we use the generalized Gell-Mann matrices as the basis for Hermitian matrices, ...
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Verify my proof that eigenvalues of a unitary matrix have absolute value 1
Show that eigenvalues of a unitary matrix have absolute value 1.
Proof below. Please verify, critique, or improve. Note: Many proofs are available; this question is to verify this proof.
Notation: $...
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What is the resulting group of $(\operatorname{U}(1)\times \operatorname{U}(1)) / \mathbb{Z}_2 $
I found this similar question to mine, however in my case I have the direct product of two copies of the group. My intuition tells me I should get $\operatorname{U}(1)$, but intuition is not proof. ...
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Embedding $D_{2n}$ in $U_2$
I have a problem doing Exercise 18.8 of groups and symmetry by M. A. Armstrong.
18.8. Look at our description of the Möbius band as a subset of $\mathbb{C} \times \mathbb{C}$ and find matrices in $...
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Assuming the violation of unitarity, what is logically possible?
In quantum mechanics you can let go of the requirement to test and measure, use instruments, have interaction with the environment, and make observations.
This applies to closed quantum systems.
In ...
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Parameterisation of a $3\times3$ unitary matrix, only knowing the squared modulus
Let's say that I have a $3\times3$ complex unitary matrix $\mathbf{M}$, with elements $a_{i,j}$.
If I only know the squared modulus of $\mathbf{M}$ and its elements (that is, I only know $|a_{i,j}|^2$)...
3
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1
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Unitary matrix corruption
Suppose I have got a unitary matrix and I want to introduce random noise to simulate data corruption. How to introduce the noise in a proper way such that the corrupted matrix is also unitary?
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Does there exist a unitary matrix such that $Uv=\|v\|e_{1}$ for $v\in\Bbb{C}^{n}$?
Let $v\in\Bbb{C}^{n}$ be a fixed vector. Then does there exist a unitary Matrix $U$ such that $Uv=\|v\|e_{1}$? or even $Uv=c\|v\|e_{1}$ for some real constant $c$?
I am looking for a Householder ...
- 1,057
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0
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Do all unitary matrices have fast multiplication algorithms?
One of the most widely known unitary matrices is the DFT matrix, which also has a well known fast multiplication algorithm $(O(n \ \textrm{log} \ n))$, the Fast Fourier Transform. Do all unitary ...
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