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 and , as described above, are groups.

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What is ?

— Me@2019-05-24 11:25:41 PM

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The set of all 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 , the first unitary group.

— Wikipedia on *Circle group*

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In mathematics, a complex square matrix is unitary if its conjugate transpose is also its inverse—that is, if

where is the identity matrix.

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

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