What is a Unitary Matrix?
A unitary matrix is a complex square matrix that, when multiplied by its conjugate transpose, results in the identity matrix. In other words, a matrix U is unitary if U*U = UU* = I, where U* is the conjugate transpose of U, and I is the identity matrix of the same dimension as U.
The concept of a unitary matrix generalizes the real orthogonal matrix to complex numbers. An orthogonal matrix is a real square matrix whose columns and rows are orthogonal unit vectors, and it holds that QTQ = QQT = I, where QT is the transpose of Q. Similarly, the columns and rows of a unitary matrix are orthonormal vectors, but in the complex vector space.
Properties of Unitary Matrices
Unitary matrices have several important properties:
- Norm Preservation: Unitary matrices preserve the Euclidean norm of vectors. If U is a unitary matrix and v is a vector, then the norm of Uv is equal to the norm of v.
- Inner Product Preservation: They also preserve inner products, which implies that angles between vectors are preserved under unitary transformations.
- Determinant: The determinant of a unitary matrix has an absolute value of 1.
- Eigenvalues: The eigenvalues of a unitary matrix lie on the unit circle in the complex plane.
- Invertibility: A unitary matrix is always invertible, and its inverse is equal to its conjugate transpose, U-1 = U*.
Conjugate Transpose
The conjugate transpose (also known as the Hermitian transpose) of a matrix is found by taking the transpose of the matrix and then taking the complex conjugate of each entry. The complex conjugate of a complex number changes the sign of the imaginary part, so for a complex number a + bi, its complex conjugate is a - bi.
Applications of Unitary Matrices
Unitary matrices are fundamental in various fields of mathematics and applied sciences:
- Quantum Mechanics: In quantum mechanics, unitary matrices are used to describe the evolution of quantum states. Since probabilities must be preserved in quantum evolution, the transformations representing these evolutions must be unitary.
- Signal Processing: The Fourier transform, which is essential in signal processing, can be represented as a unitary matrix. This property ensures that the energy of the signal is preserved when transforming between time and frequency domains.
- Control Theory: In control theory, unitary matrices are used in the analysis and design of stable control systems.
- Computer Science: In computer science, especially in the area of quantum computing, unitary operations are vital as they correspond to quantum gates and quantum circuits.
Unitary Matrix vs Orthogonal Matrix
While unitary matrices are defined over the complex field, orthogonal matrices are their real-numbered counterparts. Both types of matrices are used to represent rotations and reflections, but orthogonal matrices are limited to real vector spaces, while unitary matrices operate in complex vector spaces.
Diagonalization and Spectral Theorem
The spectral theorem states that any normal matrix, including any unitary matrix, can be diagonalized by a unitary matrix. This means that for a unitary matrix U, there exists a diagonal matrix D and a unitary matrix V such that U = VDV*. The diagonal entries of D are the eigenvalues of U, and the columns of V are the corresponding eigenvectors.
Conclusion
Unitary matrices play a crucial role in various areas of mathematics and physics due to their norm-preserving property. They are a key concept in understanding complex rotations and transformations, and their applications span from quantum mechanics to signal processing and beyond. The study of unitary matrices and their properties is an essential part of linear algebra and functional analysis.
