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 every eigenvalue to a unitary operator has absolute value 1

Let $A$ be a unitary operator on hilbert space $H$, i.e. $$(Au|Av) = (u|v)$$ for all $u,v \in D_A.$. I'm asked to show that all eigenvalues to this unitary operator has absolute value 1. My attempt: ...
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What is the meaning of the words here?

I'm not sure what the meaning of the question is, and I need some help understanding what I'm even trying to do. The unitary operator in question is in $U(4)$ The question is: "Decompose the ...
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1answer
15 views

Unitary group is a properly embedded Lie subgroup of $GL(n, \mathbb C)$

I would like to show that $U(n)$, the unitary group of degree $n$, is a properly embedded Lie subgroup of $GL(n, \mathbb C)$ of dimension $n^2$, and find the matrices that are in the tangent space at ...
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Solve $A^T\Sigma A=I$ for $A$ with $\Sigma$ being psd

Suppose $\Sigma^{p\times p}$ is positive definite. For $A^{p\times p}$, the solutions to $A^T\Sigma A=I$ can be can be given by what? Suppose $\Sigma$ has eigendecomposition $\Sigma=PLP^T$. Then $P^T\...
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Backward Stability solving linear system with unitary matrices

Let $Q \in \mathbb{C}^{n\times n}$ be a unitary matrix that can be exactly stored in floating point arithmetic. Suppose we want to solve the following linear system: \begin{equation} Qx=b \end{...
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1answer
31 views

Prove unitary evolution for an ODE

Consider a nonlinear ODE of square matrix $A(t)=(a_1(t),\cdots,a_n(t))^T$ $$\mathrm{i}\,\dot A(t) = A(t) M(t)$$ with $$M(t) = A^\dagger(t) H(t) A(t)$$ where $H(t)$ and hence $M(t)$ are Hermitian ...
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45 views

Maintain unitary time evolution for a nonlinear ODE

I want to solve a nonlinear ODE of matrix $A(t)$ $$\mathrm{i}\dot A = A(t)M(t),\:\mathrm{with}\: M(t)=A^\dagger(t)H(t)A(t)$$ where $H(t)$ and hence $M(t)$ are Hermitian. Therefore, I presume the time ...
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Cayley transforms of self-adjoint extensions

Let $\ T:{\frak Dom}(T) \rightarrow \scr H\ $ be a symmetric operator and assume $ S$ is its self-adjoint extension, that is: ${\frak Dom}(T) \subset {\frak Dom}(S)$ $S\vert_{{\frak Dom}(T)} = T$ ...
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1answer
35 views

Properties of Unitary operators on Hilbert spaces

Is it true that for every unitary extension $U' \supset U$ $\psi \in \{ {\frak Dom}(U)\}^\perp \Longleftrightarrow U'\psi \in \{ U'({\frak Dom} (U))\}^\perp$ ? My question is: how can I be sure that ...
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Forcing the determinant of a submatrix to be zero

Setup: Let $M$ be an arbitrary $4\times 4$ unitary matrix (I actually believe this conjecture is true for invertible $M$, but I am only interested in unitary). Define the matrix $$M^{-1}\left[\begin{...
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Given $U(z)=\exp(za^†_1a_2−z^∗a_1a^†2)$. Show that $U(z)^†a_1U(z)=\cos(|z|)a_1−e^{i\arg(z)}\sin(|z|)a_2.$

In my notes on quantum optics I have that a beam splitter's unitary matrix may be written as: $U(z)=\exp(za^†_1a_2−z^∗a_1a^†2)$. So $U(z)^†a_1U(z)=\cos(|z|)a_1−e^{i\arg(z)}\sin(|z|)a_2.$ How do I get ...
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How do unitary preserve inner products? Why is $\langle U x, U y \rangle = \langle x, U^\dagger U y \rangle$?

On many links online [1], [2], [3], etc. they mention the following: $U$ is a unitary matrix, i.e. $UU^\dagger = U^\dagger U = I$, where $U^\dagger$ is the Hermitian conjugate of $U$. Then, $$\langle ...
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1answer
52 views

Do singular values change when an arbitrary matrix is multiplied by a unitary matrix?

Let X be an arbitrary matrix, and let U be a unitary matrix. Is it so that the singular values of X are the same as the singular values of UX? Does the same apply for XU?
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What are the sufficient and necessary conditions for a unitary matrix to be written in a tensor product of 2x2 matrices?

I understand an arbitrary unitary matrix cannot be always written in a tensor product of 2x2 matrices. But, if a unitary matrix meets some conditions, can it be done? If so, what are those conditions?
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rank of unitary matrices

Let $U:=[u_1,\cdots,u_n] \in \mathbb{C}^{n,n}$ with $u_i \in \mathbb{C}^n$ for $i=1,\cdots,n$ be a unitary matrix. Why is $u_j u_j^H$ a rank one matrix for $j=1,\cdots,n$? ($u_j^H$ denotes the ...
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Distinction between (special) Complex Orthogonal Groups and Unitary Groups

The complex orthogonal group of complex dimension $n$ can be defined as: $$O(n;\mathbb{C})=\{M \in GL(n;\mathbb{C}) \>\> | \>\> M^TM = \mathbb{I}_n\}$$ $$SO(n;\mathbb{C})=\{M \in O(n;\...
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Partial eigendecomposition of a positive semi-definite matrix

Any positive semi-definite matrix $A$ can be decomposed into $$ A = Q \Lambda Q^\dagger, $$ where $Q$ is a unitary matrix and $\Lambda$ is a diagonal matrix. Now I would assume that there is a ...
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Invariant logarithm of a unitary matrix

We start with a finite dimensional unitary representation over $\mathbb C$ of some (compact) group $G$. In the matrix algebra, take a unitary matrix $U$ that is invariant under the conjugation action ...
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Proving unitarily equivalence using induction

Problem: A matrix B ∈ Mn,n(C) is said to be unitarily equivalent to A ∈ Mn,n(C) if there exists a unitary matrix U ∈ Mn,n(C) such that B= U†AU. Show that every matrix A ∈ Mn,n(C) is unitarily ...
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How to find unitary matrix from multiple eigenvector

enter image description here Find the unitary matrix $U$ and the upper triangular matrix $T$ such that $U^{-1}AU = T$ where $A = \pmatrix{3&2\\-2&-1}$ has the eigenvalue $\lambda = 1$ (twice) ...
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Normality result using Schur form

Let $A\in\mathbb{C}^{n\times n}$ have eigenvalues $\lambda_1,\lambda_2,\dots,\lambda_n$ Using Schur Form so that \begin{equation}\sum_{i,j=1}^{n}|a_{ij}|^2=\sum_{i=1}^{n}|\lambda_i|^2\implies A\text{ ...
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1answer
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How to get eigenvectors using QR algorithm?

From everything I've heard, this matlab code ought to spit out a matrix where each row is the same. So why doesn't it? ...
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Eigenvalues of unitary matrices over a finite field

Let $q$ be a power of an odd prime number $p$. Let us fix $\overline{\mathbb F}$ an algebraic closure of $\mathbb F_q$ and let $\sigma$ denote the Frobenius relative to $q$. Consider the group of ...
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Given $A$ is Hermitian, Show that a given $U$ is unitary

Let $A\in\mathbb{C}^{n\times n}$ be Hermitian. Show that the matrix $U=(A+iI)^{-1}(A-iI)$ is a unitary matrix. I have done so far: We want $U^*U=I$, so finding $U^*$ \begin{align}U^*&=((A+iI)^{-1}(...
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1answer
68 views

Prove that the tensor product of unitary matrices is also unitary

Show that if $A$ and $B$ are unitary matrices, then $C = A \otimes B$ is unitary.
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38 views

Parametrization of 2x2 unitary matrix with eigenvalues $\{1,-1\}$

I want to parametrize the set $\mathcal{U}_{1,-1}(2)$ of complex 2x2 unitary matrices with eigenvalues $\{1,-1\}$. I know of various parametrizations for the set $\mathcal{U}(2)$ of complex 2x2 ...
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Existence of a unitary matrix $U$

Prove that, for any complex matrix $A$ of order $3$, there exists a unitary matrix $U$ of order $3$, such that $UAU^{-1}$ is a matrix of the form $\left( \begin{matrix} *& 0& *\\ *& *&...
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1answer
41 views

Extension of isometry

The original question is Here(In polar decomposition(A=RU) decomposition, How to extend Isometry into whole complex Euclidean space) But I make brief conditions that we need. Let $X$ is complex ...
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223 views

Solutions to equation of $6$ variables

Preliminaries: Let $M$ be any $4\times 4$ unitary matrix. Now, define the matrices $A$ and $B$ by $$A=M^\dagger\left[ \begin{pmatrix} \cos(\theta) & -e^{i\lambda}\sin(\theta)\\ e^{i\phi}\sin(\...
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1answer
36 views

Proving A=RU where R is nonnegative self-adjoint mapping and U is unitary

I'm trying to following proof of this. $A$ which is linear mapping of complex Euclidean space into itself. Then $A$ can be factorized by $A=RU$ where $R$ is nonnegative self-adjoint mapping, and $U$ ...
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Existence of unitary matrix that produces quantum coin with given frequencies via iteration

I'm curious about for which $n$ there exists a $2\times 2$ unitary matrix $U(n)$ such that for $1 \le k \le n$ $$|U^k_{1,0}(n)|^2 = \frac kn,$$ where $U^k_{1,0}(n)$ is the lower left element of the $k$...
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1answer
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Claim of convexity of unitaries with $\|U\|_\infty \leq 1$ for a minimax theorem

The context of this question is the step to go from (24) to (25) here. One considers an optimization problem of the following form $$\inf_{\rho \in S}\sup_{U}\ (\cdot)$$ where $S$ is a convex set and $...
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Verifying whether a point belongs to an orbit of a finite collection of one-parameter subgroups of a unitary group $\mbox{U}(m, \mathbb{C})$

Given a finite collection of one-parameter unitary groups, $$ t \mapsto g^{t}_{k} \in \mbox{U}(m, \mathbb{C}) $$ acting on an $m$-dimensional Hermitian space $\mathbb{C}^{m}$ and having compact orbits,...
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1answer
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Using $\|V_1 V_1^\star - V_2 V_2^\star\|_\infty$ to bound $\|V_1 - V_2\|_\infty$

The converse question of the result I am asking about is here. Denote the operator norm by $\|\cdot \|_\infty$. That is $\|X\|_\infty = \max\{\|Xv\|:\|v\| = 1\}$ and $\|\cdot\|$ is the Euclidean 2-...
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1answer
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Bound for $\|U_1XU_1^\star- U_2XU_2^\star\|_\infty$ for unitary $U_i$

Denote the operator norm by $\|\cdot \|_\infty$. That is $\|X\|_\infty = \max\{\|Xv\|:\|v\| = 1\}$ and $\|\cdot\|$ is the Euclidean 2-norm for vectors. Let $U_1, U_2$ be unitary matrices (i.e. $U_i^\...
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Given the unitary matrix $X = ABCD$, is there a way to recover $DCBA$ only using operations on $X$?

Suppose I have square unitary matrices $A, B, C, D$. I also have the unitary matrix $X = ABCD$ (or more generally, $X = X_1X_2\cdots X_n$). Is there a way to recover $Y=DCBA$ (or more generally, $Y = ...
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How to prove that the spectral norm is unitarily invariant?

How to prove that $\| U A \|_2 = \| A U \|_2 = \| A \|_2$ for any unitary matrix $U$? It can be proved that $$\| UA\|_2 = \sqrt{\lambda_{\max}({(UA)}^*(UA))} = \sqrt{\lambda_{\max}(A^*U^*UA)} = \sqrt{\...
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Linear algebra unitary operator problem

I am not undertanding what I have to do in letter (c) of this problem.. I just wrote $\alpha_1=\begin{bmatrix} \cos\phi & -\sin\phi\\ \sin\phi & \cos\phi \end{bmatrix} \begin{bmatrix}1\\0\end{...
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1answer
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Is $0 \rightarrow SU(n) \rightarrow SU(n+1) \rightarrow S^{2n + 1} \rightarrow 0$ exact?

I recently went to a talk where the speaker mentioned that $$ 0 \rightarrow SU(n) \rightarrow SU(n+1) \rightarrow S^{2n + 1} \rightarrow 0$$ was exact. I think it's well-known that this sequence is a ...
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Is the function representation $T_{\infty} : U(1) → GL(L^2(S^1))$ unitary?

Let $S^1$ be a $U(1)$-set, and let $L^2(S^1)$ be the Hilbert space of square-integrable complex-valued functions $f : S^1 → \Bbb C$. It can be shown that the map $$ T_{\infty} : U(1) → \text{GL}(L^2(S^...
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Construction of QR decomposition for a singular matrix

Consider a matrix $A \in \mathbb{C}^{n \times n}$. Suppose it's non-singular. Then it's columns $a_1, a_2, ..., a_n$ form a basis in $\mathbb{C}^n$. Let's apply Gram-Schmidt process to them. We will ...
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1answer
56 views

Proof of the Cayley Transform. Showing that $H=i(I-U)(I+U)^{-1}$ where $U=(I+iH)(I-iH)^{-1}$ for a Hermitian matrix $H$.

I am working on showing that a Hermitian matrix $H=i(I-U)(I+U)^{-1}$ where $U=(I+iH)(I-iH)^{-1}$. I have already shown that $I-iH$ is invertible and that $U$ is unitary, but am stuck on showing that $...
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1answer
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Proof of the matrix form of the Lie algebra of $SU(2)$

The Lie algebra elements fulfill $$\mathfrak su(2) =\{X: \text {Tr} (X) = 0 \;, X^\dagger = - X \}.\tag 1$$ $$\mathfrak su(2) = \left\{\begin{bmatrix} ix & y + iz \\ -y+iz & -ix \end{bmatrix}\;...
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1answer
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how to encode a unitary matrix using a set of **independent** variables?

I am working on an optimization problem related to a unitary matrix $X$, and wonder whether it is possible to encode a unitary matrix using a set of independent variables? For instance, for a 2D ...
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Generating representations of U(n) using Schur functions.

We know that Schur functions $S_\lambda$ are related with irreps. of $U(n)$ and that there is an associated branching rule for the subgroup chain $U(1)\subseteq \ldots \subseteq U(n-1)\subseteq U(n)$, ...
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$U(n)$ is a compact group. Proof.

Consider a unitary group $U(n)=\{A\in GL_n(\mathbb{C}):A^*A=I\}$. I want to show that $U(n)$ is a compact group. First, I can observe that $U(n)\subseteq M_n(\mathbb{C})$ i.e. it's a subspace of an ...
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41 views

Understanding a proof that “the columns of a unitary matrix are orthonormal”

Goal: Let $u_i$ and $u_j$ be the $i$th and $j$th columns of unitary matrix $U$, respectively. We wish to show that $$ \langle u_i, u_j \rangle = 0, i \ne j \\ \langle u_i, u_j \rangle = 1, i = j \\ $$...
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1answer
36 views

Can we solve a matrix equation when the vectors are given and the matrix is variable?

Usually, a matrix equation means that $$ Ax = b $$ when A and b are given and x is the variable we want to know. However, when x and b are given and we want to know the value of the matrix, is it ...
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1answer
26 views

Unitary transformation to make diagonal elements zero?

For arbitrary $A\in\mathbb{C}^{3\times 3}$, is it always possible to find a unitary matrix $Q$ such that $QA$ has zero diagonals? Namely, to make $QA$ in the following form: $$ QA=\begin{bmatrix} ...
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Prove that A is zero matrix

Let $A$ be an $n×n $ complex matrix such that the three matrices $A+I$ , $A^2+I $ , $ A^3+I$ are all unitary .Prove that$ A$ is the zero matrix I try to show that $Trace( A^{\theta}A) =0$ where $A^{\...

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