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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Relationship between a vector and its discrete fourier transform

Suppose we have a real vector $a=(a_0,\ldots,a_{n-1})$ and let $b=(b_0,\ldots,b_{n-1})$ be its discrete fourier trasnform. So $$\mathscr{F}(a_k)=b_k=\sum_{i=0}^{n-1}{a_i}e^{\frac{-j2\pi i k}{n}}$$Let $...
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Prove that $T = cU$, where c is a scalar and $U$ is unitary.

In $\mathbb{C}^n$, with inner product $\langle z,w \rangle = \sum_{j} z_j\overline{w}_j$. Let $T$ a linear operator such that $\langle T(z), T(w) \rangle = 0$, if $\langle z,w \rangle = 0$. Prove ...
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How to prove: If $A$ and $B$ are normal matrices and AB = BA, $A+B$ and $AB$ are also normal, and $A$ and $B$ are simultaneously diagonalizable.

If $A$ and $B$ are normal matrices and they commute $(AB=BA)$, then: $A+B $ is normal $AB $ is normal $A$ and $B$ are simultaneously diagonalizable: there is a unitary matrix $U$ such that both $U^*...
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Polar decomposition of a linear combination of unitary matrices

Consider a complex-valued square matrix $M$ of the form $$M = \frac{1}{2}\left(U_1 + e^{-i\phi}U_2\right),$$ where $U_1$ and $U_2$ are unitary matrices and $\phi$ is a real number. Moreover, consider ...
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Matrix of change of basis between orthonormal bases is unitary

Question statement: Let $V$ be a vector space over $\mathbb{C}$ with an inner product $\langle,\rangle$. Assume that $\mathcal{A}=\{w_1, ..., w_n\}$ and $\mathcal{B}=\{v_1,...,v_n\}$ are two ...
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Unitarily diagonalizable V.S. unitary matrix

We work over $\mathbb{C}$. Let $A\in M_n(\mathbb{C})$. (so our vector space is finite dimensional.) We know that $A$ is unitary iff $AA^*=I=A^*A$ where $^*$ is conjugate transpose. $A$ is normal iff ...
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Probability density function invariant under unitary transformation

Recently I have been studying Mehta's book on Random Matrices (3rd edition). In this book the author defines the Gaussian Unitary Ensemble in the set of hermitian matrices with 2 specific properties. ...
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a problem about unitarily similarity on "Linear Algebra" by Kenneth Hoffman and Ray Kunze Sec.9.5 Q.8

Let $V$ be the space of complex $n$×$n$ matrices equipped with the inner product $\langle A,B\rangle=\operatorname{tr}(AB^*)$.Then let $L_B$,$R_B$ denote the linear operators on $V$ defined by $L_B(A)=...
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What is $ d B_t d t^{1/2} $, where $B_t$ is the Brownian motion on $\mathbb{R}$?

Consider the time-evolution operator $U(dt) = \exp(-id tJH_0 - i d B_t V)$ where $H_0$ and $V$ are some Hamiltonians, and $J$ is some coupling. $B_t$ is the Brownian motion on $\mathbb{R}$ with $B_0 = ...
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Determinant inequality with unitary matrix

I come up with the following conjecture while doing my research, which is a determinant inequality. I have tried to run the MatLab simulation to verify its sanity. It seems that the inequality is true....
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Unitary operator times projection operator

In this paper, the authors claim that for $C$ a unitary operator and $P$ a projection operator, if $CP \propto P$, then the constant of proportionality must be one. I don't see why this must be the ...
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Property of unitary matrices

This article - https://doi.org/10.1016/0370-2693(88)91216-6 - states that for a $3 \times 3$ unitary matrix $A$, we have $|A_{33}| = |(A_{11}A_{22}-A_{12} A_{21})|$, where $A_{ij}$ stands for the ...
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If $A$ is positive semidefinite and $P$ is unitary, is $B = P^{-1}AP$ positive semidefinite?

I know that in general, positive semidefiniteness is not preserved by matrix similarity. But is it preserved when $B = P^{-1}AP$ and $P$ is unitary?
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Are submatrices of an arbitrary complex unitary matrix diagonalizable?

Consider a complex unitary matrix $U \in \mathbb{C}^{N\times N}$ and pick two diagonal and two off-diagonal elements from its $m$th and $n$th rows to construct a $2 \times 2$ submatrix: \begin{...
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Can any unitary matrix be written as a product of "2D" unitary matrices?

Given a $n\times n$ unitary matrix $\mathbf{U}$, can this be rewritten as a product $$ \mathbf{U}=\prod_{i=1}^n\mathbf{U}_i $$ where $\mathbf{U}_i$ are unitary matrices themselves, but they only ...
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Show that $H \mathrel{\unlhd} SU(2)$ contains matrices with all possible traces between $2$ and $2\cos \varphi_0 $.

Let $H \mathrel{\unlhd} SU(2)$ that contains an element $A$ such that $\text{tr}(A)=2\cos \varphi_0 \neq \pm2$. Use i),ii) to show that $H \mathrel{\unlhd} SU(2)$ contains matrices with all possible ...
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Correlations in implementations of random unitary channels

A random unitary channel (RUC) acting on a quantum system $S$ is any completely positive and trace preserving (CPTP) linear map $\mathcal{E}$ that can be expressed as \begin{align} \mathcal{E}(\...
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Is $SU(n)$ a complex Lie group? [duplicate]

I had been naively thinking that $GL_n(\mathbb C)$,$SL_n(\mathbb C)$,$U_n(\mathbb C)$ and $SU_n(\mathbb C)$ being the complex analogs of $GL_n(\mathbb R)$ etc should be complex Lie groups. But wiki ...
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Query regarding commuting family of unitary operators on a finite dimensional complex inner product space

We are asked to prove- If $\{T_i\}$ is a commuting family of unitary operators over a finite dimensional complex inner product space $V$. Then $$V=\oplus_{\lambda} V_\lambda$$ where each $V_\lambda $...
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Decomposing a unitary matrix into block unitary matrices

Can a square non-zero and non-diagonal $n\times n$ complex unitary matrix be decomposed into non-zero and non-diagonal square $m\times m$ complex unitary matrices (such that $n > m$)? For example ...
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Quantum gate for inner product with given vector

Let $\mathcal{H}$ be a space of $n$-qudits over $\mathbb{F}_d$, i.e. $(\mathbb{C}^d)^{\otimes n}$. And I have a given vector $a=(a_1,\dots,a_n)\in\mathbb{F}_d^n$. I wish to construct an inner product ...
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SU(2) invariant structures and the Frobenius Schur indicator

For $SU(2)$, the fundamental representation is a quaternionic representation. Which means there is a preserved skew symmetric form, written as a matrix: $$ \varepsilon = \left(\begin{array}{cc} 0 &...
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Is there a name for this 'splitting' of an orthogonal structure into unitary and symplectic structures?

Suppose we have a (non-trivial) representation of some special orthogonal group $SO(p,q)$ over a real vector space $V$, I.e. the action of elements of $SO$ leave invariant a non-degenerate symmetric ...
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Unitary transformation with T invariant subspace

I encountered the following problem and I was wondering whether it is possible to deduce even stronger conclusion that $T = \pm I$ This is the problem: Let $T: V \to V$ be a linear transformation on ...
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Matrices that are simultaneously Hermitian and unitary [duplicate]

My quantum mechanics professor was discussing the properties of Pauli matrices, their being both Hermitian and unitary. Then he made a remark that it is not possible to find three $n \times n$ ...
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Unitary equivalence of sums of unitary equivalent hermitian matrices

Consider two hermitian matrices $A$ and $B$. Suppose that there exists a unitary matrix $U$ such that $A+B$ is unitarily equivalent to $U A U^* +B$. Does this imply that there exists a unitary matrix $...
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Number of solutions in a finite field corresponding to one-dimensional isotropic subspaces

Let $q$ be a prime power and define the bilinear form $$((x_1, x_2, x_3),(y_1, y_2, y_3)) \mapsto x_1 y_3^q + x_2 y_2^q +x_3 y_1^q$$ on the three-dimensional vector space $V$ over the finite field $F_{...
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Unitary representations have unitary matrices

I want to show that for each unitary representation $\pi: G \to GL(\mathbb{C}^n)$, the matrix $A \in GL_n(\mathbb{C})$ "corresponding" to $\pi$ is also unitary. So I suppose $\pi$ is ...
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1 vote
2 answers
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Prove unitary matrices do not alter the magnitude of a column vector with complex elements

Prove the following: Given $r'=Ur$ with $U$ a unitary matrix and $r$ a (column) vector with complex elements, show that the magnitude of $r$ is the same as the magnitude of $r'$. The matrix $U$ ...
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Can a Hankel matrix $H$ be efficiently decomposed into a linear combination of unitaries (LCU), so that $H=\sum_k a_k U_k$

Suppose I have a Hankel matrix of arbitrary size $N\times M=2^n\times 2^m$ for integers $n<m$, given by: $H=\begin{pmatrix}x_1&x_2&\cdots & x_M\\x_2&x_3&\cdots& x_1\\\vdots&...
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Matrix-rank nonincreasing unitary tensor operations

I have a multidimensional array $A_{ijkl}$ $\in\mathbb{C}^{m\times n\times o \times p}$ indexed by four integers $i,j,k,l$. I will call $i$ and $j$ the "left" indices, $j$ and $k$ the "...
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-1 votes
1 answer
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Unitary transformation on vector is equal to the transpose on its orthogonal vector?

While working on my Quantum course, I have observed that U|0> = UT|1> for a Unitary matrix U. This is solely based on observations and calculating the matrices. I wish to try and prove this ...
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1 answer
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Show that no $d×d$ unitary matrix $U$ satisfies the condition $|\langle \psi | U | \psi \rangle|^2 = 0$ for every $|\psi_i| \in C^d$

Show that no $d×d$ unitary matrix $U$ satisfies the condition $|\langle \psi | U | \psi \rangle|^2 = 0$ for every $|\psi_i\rangle \in C^d$. How would I show this? So far, I've observed that it's ...
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1 answer
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Tensor product of groups and irreducible representations

Let $ A $ be a finite subgroup of $ SU_{n_1} $ such that the inclusion $ A \hookrightarrow SU_{n_1} $ is an irrep. And let $ B $ be a finite subgroup of $ SU_{n_2} $ such that the inclusion $ b \...
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2 votes
1 answer
69 views

Making complex matrix real with unitary transformation

Consider the Hermitian matrices: $$A=\begin{bmatrix} 0 & -0.7011-0.0912i & 0 \\ -0.7011+0.0912i & 0 & 0.6702-0.2255i\\ 0 & 0.6702+0.2255i & 0 \end{bmatrix}\\ B=...
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4 votes
1 answer
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Volume of SU(2) and Haar Integral

The Pauli matrices are given by $$\sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma_3 = \begin{...
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Computing orthogonal matrix based on existing orthogonal matrix

If I only have an orthogonal rectangular matrix Q of dimensions m x n, how can I generate an orthogonal matrix Q' of dimensions m x (n+1), that will have the same n first columns as Q? Is it possible ...
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If 4 Unitary maps amount to the identity, whats the general relationship between them?

Say we play a game where we pick a random point $\rho \in \mathcal{B}(\mathcal{H})$. The rules are, you can pick any 4 unitary $(\mathcal{U}:\mathcal{B}(\mathcal{H}) \rightarrow \mathcal{B}(\mathcal{H}...
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One-parameter subgroups of $GL_n(\mathbb{C})$ passing through a given real matrix

Let $\gamma$ be a matrix in $GL_n(\mathbb{R})$. Then $\gamma$ lies on a 1-parameter subgroup of $GL_n(\mathbb{C})$ even though it may not lie on a 1-parameter subgroup of $GL_n(\mathbb{R})$ (for ...
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Number of free real parameters in Hermitian vs unitary matrix

There seems consensus that the number of free real parameters of an $n \times n$ Hermitian matrix $M$ is $n^2$; see e.g. this post and also this short note. On the other hand, such a matrix has a ...
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0 votes
1 answer
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Properties of matrix $A$ when $D^H A^{-1} D = D^H D (D^H A D)^{-1} D^H D$

Let $\mathbf{A} \in \mathbb{C}^{N \times N}$ be a Hermitian matrix and let $\mathbf{D} \in \mathbb{C}^{N \times K}$ (with $K<N$) be a semi-unitary matrix such that $\mathbf{D}^{H} \mathbf{D} = \...
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2 answers
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Unitarity similarity transformation

Prove that $R$ and $R^{\dagger}$ can be diagonalized by a common unitary similarity transformation if $R^{\dagger}$ is commutable with $R$. Let $R = SMS^{-1}$, where $M$ is diagonal and $S$ is ...
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Unitary matrices U, U* that equalise the elements on the main diagonal [closed]

Can I find unitary matrices U, U* that can equalise the elements on the main diagonal of a given matrix M? $$A=U^* M U,$$ such that $A_{11}=A_{22}=A_{33}=...$
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1 answer
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A and B have identical singular-values and identical eigenvalues, are they unitary similar?

I've come across this question while taking a matrix analysis course. Given: $A,B \in\mathbb{C}^{3 \times 3}$ with $\lambda_1(A) = \lambda_1(B) ,\,\lambda_2(A) = \lambda_2(B),\,\lambda_3(A) = \...
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0 votes
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What's the probability distribution for a single entry of a random unitary (or orthogonal) matrix?

Consider a random $n \times n$ unitary matrix, i.e. whose probability measure is the Haar measure for $\mathrm{U}(n)$. What is the marginal probability distribution for a single element of this matrix?...
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Difference between $SO_n(\mathbb{R})$ and $SU_n(\mathbb{R})$.

Is there any difference between the real Lie groups $SO_n(\mathbb{R})$ and $SU_n(\mathbb{R})$ or similarly the Lie groups $O_n(\mathbb{R})$ and $U_n(\mathbb{R})$. It seems that usually when talking ...
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2 votes
0 answers
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Is the truncated Frobenius norm convex or not?

Given a positive integer $k$ and a matrix $X\in \mathbb{R}^{m\times n}$. A truncated Frobenius norm of a matrix $X$ is defined by $$\Vert X \Vert_{k,F} = \sqrt{\sum_{i=k+1}^{m} \sigma_i^2(X)},$$ where ...
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4 votes
0 answers
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A question on irreducible unitary representations of the unitary group $ U(n)$

Let $n$ be a positive integer and $U(n)$ be the group of $n\times n$ unitary matrices. I have two question regarding the irreducible unitary representations of $U(n)$. Is there any irreducible ...
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1 answer
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If $A_1$ and $A_2$ are unitary matrices ,and if $CA_1C^{-1}=A_2$,then is $C$ is also unitary?

$A_1$ and $A_2$ are matrix of the same unitary transformation, w.r.t different orthonormal basis. And $C$ is the transition matrix(hence, $CA_1C^{-1}=A_2$). To show if $A_1$ and $A_2$ are unitary ...
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Proof that the determinant of the Hermitian conjugate of $A$ is the complex conjugate of the determinant of $A$

While revising for an exam, I came across the following proof that the determinant of a unitary matrix $A$ is $\pm 1$: $$1=\det(I)=\det(A^{\dagger}A) = \det(A)^{*}\det(A)=|\det(A)|^2 $$ This seems to ...
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