Question

Is it compulsory that a unitary matrix should contain complex numbers having an imaginary part?Can a matrix be unitary even if its elements have imaginary part 0 and satisfies P.P* =I? Can we call a unit matrix unitary?

Answer 1

Answer:

In linear algebra, a complex square matrix U is unitary if its conjugate transpose U* is also its inverse, that is, if

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

where I is the identity matrix.

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

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