# 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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### 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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### 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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1 vote
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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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