# 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-...
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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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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 $\tau \in ... 1 vote 0 answers 43 views ### Finitely generated dense subgroup of an unitary group I am interested in finitely generating dense subgroups of the unitary group$U(d)$of order$d$(with respect to the operator norm topology), and am considering the following question. Let$I_d$be ... 2 votes 0 answers 38 views ### Triple Product Formula on$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} & \... 1 vote 0 answers 26 views ### 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 ... 0 votes 1 answer 80 views ### 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. ... 0 votes 0 answers 48 views ### 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 ... 0 votes 0 answers 66 views ### 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:... 2 votes 1 answer 108 views ### 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 \... 0 votes 0 answers 45 views ### 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 &... 0 votes 0 answers 36 views ### 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}$$always hold for some unitary \boldsymbol{\... 0 votes 0 answers 18 views ### circulant Schur decomposition Let A_{1},A_{2},\dots,A_{m} be arbitrary n\times n complex matrices. Prove that there are n\times n unitary matrices Q_{1},Q_{2},\dots,Q_{m} such that matrices$$Q_{1}^{*}A_{1}Q_{2},\ Q_{2}^{*... 1 vote 1 answer 41 views ### 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 $$W(T)=\{\langle Tx,x\rangle |x\in H,\|x\|=1\}$$... 0 votes 0 answers 15 views ### 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 ... 1 vote 2 answers 72 views ### 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 $$(\overline{A})_{ij} := \overline{a_{... 0 votes 0 answers 15 views ### Matrix multiplication involving a unitary matrix in exponential form and a simple matrix I am not sure if this is even a valid question, but it will help to clear my doubts anyway. For context, check out Violating the Thermodynamic Uncertainty Relation in the Three-Level Maser. So I am ... 0 votes 0 answers 37 views ### Under what conditions does B = A B A^\intercal imply A = I for a symmetric matrix B [duplicate] I'm not sure this is generically true, but as part of something I'd like to prove I'm hoping to make use of something along the lines of being able to write for some symmetric, nonzero B$$ B = A B ... 1 vote 0 answers 22 views ### 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$be the image ... 1 vote 1 answer 19 views ### Modified orthogonal relation between unitary matrices with columns multiplied by a complex factor Suppose$U$is a unitary matrix of dimensions$n\times n$; then we know ($U^{\dagger}$is conjugate transpose): \begin{eqnarray} \sum_k U_{nk}U_{km}^{\dagger}=\delta_{nm}. \end{eqnarray} How does this ... 0 votes 1 answer 42 views ### SVD of product of diagonal and unitary matrices Given two (possibly rectangular) diagonal matrices$\Sigma_\text{L}$and$\Sigma_\text{R}$with nonnegative elements, what can we say about the singular value decomposition of $$\Sigma_\text{L} X \... 1 vote 1 answer 45 views ### su(2) vs ZXZ decomposition I know that, if we consider$$e^{i \alpha Z} e^{i \beta X} e^{i \gamma Z}$$(where X, Z are the Pauli matrices) then we can get any element of SU(2) (the so-called ZXZ decomposition). If I write ... 1 vote 1 answer 58 views ### Existence of unitary operator affecting partial trace? [closed] Let \rho and \phi be any two different density matrices on H_A \otimes H_B such that Tr_B (\rho) = Tr_B (\phi). Does there always exist a unitary U on H_A \otimes H_B, U\rho U^{\dagger} = ... 0 votes 1 answer 28 views ### A formula for a unitary matrix Proposition: U = (H+iI)(H-iI)^-1 is unitary if H is self-adjoint. I'm having trouble finding a proof. It's straightforward if (H-iI) and (H+iI)^-1 commute. But I don't see it. 0 votes 0 answers 89 views ### Exercise 4.9 Nielsen and Chuang's "Quantum Computation and Quantum Information I am trying to work through an online solution to Exercise 4.9 in Nielsen and Chuang's "Quantum Computation and Quantum Information: The question is: Explain why any single qubit unitary operator ... 5 votes 0 answers 84 views ### Prove that set of matrices is dense in U(2) Consider the group of matrices B generated by taking products of the matrices \begin{equation} \rho_1 = \begin{pmatrix}\exp(-4\pi i/5) & 0\\ 0 & \exp(3\pi i/5)\end{pmatrix}\\ \rho_2 = \begin{... 2 votes 0 answers 74 views ### Implication of mean ergodic theorem My doubt is the following. Mean ergodic theorem tells us that (there are several versions of the theorem, I will write it down as involving unitary operators on a Hilbert space just because it is ... 1 vote 0 answers 75 views ### What is the geometric meaning of the condition det(A) = 1 for A \in SU(n)? Let A be a real square matrix. The condition that A preserves the inner product, and hence preserves lengths, is the interpretation of A \in O(n). If we further require that det(A) = 1, then ... 1 vote 0 answers 32 views ### Generators of SU(n) The generators of SU(N) have the following properties: Trace[T^{a}T^{b}]=\frac{1}{2} \delta^{ab}---(i) and Trace[T^{a}]=0---(ii). In product form$$T^{a}T^{b}=\frac{1}{2}(\frac{1}{N} \delta^{ab} ... 0 votes 0 answers 47 views ### 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=... 112 views

### 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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### 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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### 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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### 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 $|a_{i,j}|^2$)...
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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 ...
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 ...