# 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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13 views

### 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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### 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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### 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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### 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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### 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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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:$$\begin{bmatrix} i &... 1answer 85 views ### Verify the matrix exponential e^{i\hat{H}t/\hbar} is unitary What could be an example of an anti-hermitian matrix i\hat{H}t , which satisfies the matrix exponential$$e^{i\hat{H}t/\hbar}$$being a unitary matrix? 1answer 45 views ### Proof that \dfrac{\partial U_t}{\partial t} U_t^\dagger is anti-Hermitian by integration by parts [closed] A proof to understand why$$\frac{\partial U_t}{\partial t} U_t^\dagger is anti-Hermitian.
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 = \...