- Source: Unitary matrix
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
U
∗
U
=
U
U
∗
=
I
,
{\displaystyle U^{*}U=UU^{*}=I,}
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
U
†
U
=
U
U
†
=
I
.
{\displaystyle U^{\dagger }U=UU^{\dagger }=I.}
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.
Properties
For any unitary matrix U of finite size, the following hold:
Given two complex vectors x and y, multiplication by U preserves their inner product; that is, ⟨Ux, Uy⟩ = ⟨x, y⟩.
U is normal (
U
∗
U
=
U
U
∗
{\displaystyle U^{*}U=UU^{*}}
).
U is diagonalizable; that is, U is unitarily similar to a diagonal matrix, as a consequence of the spectral theorem. Thus, U has a decomposition of the form
U
=
V
D
V
∗
,
{\displaystyle U=VDV^{*},}
where V is unitary, and D is diagonal and unitary.
|
det
(
U
)
|
=
1
{\displaystyle \left|\det(U)\right|=1}
. That is,
det
(
U
)
{\displaystyle \det(U)}
will be on the unit circle of the complex plane.
Its eigenspaces are orthogonal.
U can be written as U = eiH, where e indicates the matrix exponential, i is the imaginary unit, and H is a Hermitian matrix.
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.
Equivalent conditions
If U is a square, complex matrix, then the following conditions are equivalent:
U
{\displaystyle U}
is unitary.
U
∗
{\displaystyle U^{*}}
is unitary.
U
{\displaystyle U}
is invertible with
U
−
1
=
U
∗
{\displaystyle U^{-1}=U^{*}}
.
The columns of
U
{\displaystyle U}
form an orthonormal basis of
C
n
{\displaystyle \mathbb {C} ^{n}}
with respect to the usual inner product. In other words,
U
∗
U
=
I
{\displaystyle U^{*}U=I}
.
The rows of
U
{\displaystyle U}
form an orthonormal basis of
C
n
{\displaystyle \mathbb {C} ^{n}}
with respect to the usual inner product. In other words,
U
U
∗
=
I
{\displaystyle UU^{*}=I}
.
U
{\displaystyle U}
is an isometry with respect to the usual norm. That is,
‖
U
x
‖
2
=
‖
x
‖
2
{\displaystyle \|Ux\|_{2}=\|x\|_{2}}
for all
x
∈
C
n
{\displaystyle x\in \mathbb {C} ^{n}}
, where
‖
x
‖
2
=
∑
i
=
1
n
|
x
i
|
2
{\textstyle \|x\|_{2}={\sqrt {\sum _{i=1}^{n}|x_{i}|^{2}}}}
.
U
{\displaystyle U}
is a normal matrix (equivalently, there is an orthonormal basis formed by eigenvectors of
U
{\displaystyle U}
) with eigenvalues lying on the unit circle.
Elementary constructions
= 2 × 2 unitary matrix
=One general expression of a 2 × 2 unitary matrix is
U
=
[
a
b
−
e
i
φ
b
∗
e
i
φ
a
∗
]
,
|
a
|
2
+
|
b
|
2
=
1
,
{\displaystyle U={\begin{bmatrix}a&b\\-e^{i\varphi }b^{*}&e^{i\varphi }a^{*}\\\end{bmatrix}},\qquad \left|a\right|^{2}+\left|b\right|^{2}=1\ ,}
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
det
(
U
)
=
e
i
φ
.
{\displaystyle \det(U)=e^{i\varphi }~.}
The sub-group of those elements
U
{\displaystyle \ U\ }
with
det
(
U
)
=
1
{\displaystyle \ \det(U)=1\ }
is called the special unitary group SU(2).
Among several alternative forms, the matrix U can be written in this form:
U
=
e
i
φ
/
2
[
e
i
α
cos
θ
e
i
β
sin
θ
−
e
−
i
β
sin
θ
e
−
i
α
cos
θ
]
,
{\displaystyle \ U=e^{i\varphi /2}{\begin{bmatrix}e^{i\alpha }\cos \theta &e^{i\beta }\sin \theta \\-e^{-i\beta }\sin \theta &e^{-i\alpha }\cos \theta \\\end{bmatrix}}\ ,}
where
e
i
α
cos
θ
=
a
{\displaystyle \ e^{i\alpha }\cos \theta =a\ }
and
e
i
β
sin
θ
=
b
,
{\displaystyle \ e^{i\beta }\sin \theta =b\ ,}
above, and the angles
φ
,
α
,
β
,
θ
{\displaystyle \ \varphi ,\alpha ,\beta ,\theta \ }
can take any values.
By introducing
α
=
ψ
+
δ
{\displaystyle \ \alpha =\psi +\delta \ }
and
β
=
ψ
−
δ
,
{\displaystyle \ \beta =\psi -\delta \ ,}
has the following factorization:
U
=
e
i
φ
/
2
[
e
i
ψ
0
0
e
−
i
ψ
]
[
cos
θ
sin
θ
−
sin
θ
cos
θ
]
[
e
i
δ
0
0
e
−
i
δ
]
.
{\displaystyle U=e^{i\varphi /2}{\begin{bmatrix}e^{i\psi }&0\\0&e^{-i\psi }\end{bmatrix}}{\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \\\end{bmatrix}}{\begin{bmatrix}e^{i\delta }&0\\0&e^{-i\delta }\end{bmatrix}}~.}
This expression highlights the relation between 2 × 2 unitary matrices and 2 × 2 orthogonal matrices of angle θ.
Another factorization is
U
=
[
cos
ρ
−
sin
ρ
sin
ρ
cos
ρ
]
[
e
i
ξ
0
0
e
i
ζ
]
[
cos
σ
sin
σ
−
sin
σ
cos
σ
]
.
{\displaystyle U={\begin{bmatrix}\cos \rho &-\sin \rho \\\sin \rho &\;\cos \rho \\\end{bmatrix}}{\begin{bmatrix}e^{i\xi }&0\\0&e^{i\zeta }\end{bmatrix}}{\begin{bmatrix}\;\cos \sigma &\sin \sigma \\-\sin \sigma &\cos \sigma \\\end{bmatrix}}~.}
Many other factorizations of a unitary matrix in basic matrices are possible.
See also
References
External links
Weisstein, Eric W. "Unitary Matrix". MathWorld. Todd Rowland.
Ivanova, O. A. (2001) [1994], "Unitary matrix", Encyclopedia of Mathematics, EMS Press
"Show that the eigenvalues of a unitary matrix have modulus 1". Stack Exchange. March 28, 2016.
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- Matriks uniter
- Nilai dan vektor eigen
- Unitary matrix
- Normal matrix
- Hermitian matrix
- Symmetric matrix
- Unitary group
- Matrix decomposition
- Polar decomposition
- Singular value decomposition
- Quantum logic gate
- Matrix representation
