In linear algebra, an invertible complex square matrix U is unitary if its matrix inverse U−1 equals its conjugate transpose U*, that is, if
where I is the identity matrix.
In physics, especially in quantum mechanics, the conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted by a dagger (†), so the equation above is written
A complex matrix U is special unitary if it is unitary and its matrix determinant equals 1.
For real numbers, the 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.
For any unitary matrix U of finite size, the following hold:
For any nonnegative integer n, the set of all n × n unitary matrices with matrix multiplication forms a group, called the unitary group U(n).
Every square matrix with unit Euclidean norm is the average of two unitary matrices. [1]
If U is a square, complex matrix, then the following conditions are equivalent: [2]
One general expression of a 2 × 2 unitary matrix is
which depends on 4 real parameters (the phase of a, the phase of b, the relative magnitude between a and b, and the angle φ). The form is configured so the determinant of such a matrix is
The sub-group of those elements with is called the special unitary group SU(2).
Among several alternative forms, the matrix U can be written in this form:
where and above, and the angles can take any values.
By introducing and has the following factorization:
This expression highlights the relation between 2 × 2 unitary matrices and 2 × 2 orthogonal matrices of angle θ.
Another factorization is [3]
Many other factorizations of a unitary matrix in basic matrices are possible. [4] [5] [6] [7] [8] [9]
The physics of large systems is often understood as the outcome of the local operations among its components. Now, it is shown that this picture may be incomplete in quantum systems whose interactions are constrained by symmetries.
In linear algebra, an invertible complex square matrix U is unitary if its matrix inverse U−1 equals its conjugate transpose U*, that is, if
where I is the identity matrix.
In physics, especially in quantum mechanics, the conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted by a dagger (†), so the equation above is written
A complex matrix U is special unitary if it is unitary and its matrix determinant equals 1.
For real numbers, the 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.
For any unitary matrix U of finite size, the following hold:
For any nonnegative integer n, the set of all n × n unitary matrices with matrix multiplication forms a group, called the unitary group U(n).
Every square matrix with unit Euclidean norm is the average of two unitary matrices. [1]
If U is a square, complex matrix, then the following conditions are equivalent: [2]
One general expression of a 2 × 2 unitary matrix is
which depends on 4 real parameters (the phase of a, the phase of b, the relative magnitude between a and b, and the angle φ). The form is configured so the determinant of such a matrix is
The sub-group of those elements with is called the special unitary group SU(2).
Among several alternative forms, the matrix U can be written in this form:
where and above, and the angles can take any values.
By introducing and has the following factorization:
This expression highlights the relation between 2 × 2 unitary matrices and 2 × 2 orthogonal matrices of angle θ.
Another factorization is [3]
Many other factorizations of a unitary matrix in basic matrices are possible. [4] [5] [6] [7] [8] [9]
The physics of large systems is often understood as the outcome of the local operations among its components. Now, it is shown that this picture may be incomplete in quantum systems whose interactions are constrained by symmetries.