# Quick Calculation 15.1

A First Course in String Theory

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Recall that a group is a set which is closed under an associative multiplication; it contains an identity element, and each element has a multiplicative inverse. Verify that $\displaystyle{U(1)}$ and $\displaystyle{U(N)}$, as described above, are groups.

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What is $\displaystyle{U(1)}$?

— Me@2019-05-24 11:25:41 PM

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The set of all $\displaystyle{1 \times 1}$ unitary matrices clearly coincides with the circle group; the unitary condition is equivalent to the condition that its element have absolute value 1. Therefore, the circle group is canonically isomorphic to $\displaystyle{U(1)}$, the first unitary group.

— Wikipedia on Circle group

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In mathematics, a complex square matrix $\displaystyle{U}$ is unitary if its conjugate transpose $\displaystyle{U^*}$ is also its inverse—that is, if

$\displaystyle{U^{*}U=UU^{*}=I,}$

where $\displaystyle{I}$ is the identity matrix.

In physics, especially in quantum mechanics, the Hermitian conjugate of a matrix is denoted by a dagger ($\displaystyle{\dagger}$) and the equation above becomes

$\displaystyle{U^{\dagger }U=UU^{\dagger }=I.}$

The real analogue of a unitary matrix is an orthogonal matrix. Unitary matrices have significant importance in quantum mechanics because they preserve norms, and thus, probability amplitudes.

— Wikipedia on Unitary matrix

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2019.05.25 Saturday ACHK