Unitary matrix
Definition
A Unitary matrix is a type of square matrix that satisfies the condition that the conjugate transpose of the matrix is equal to its inverse. This can be represented mathematically as U*U = UU* = I, where U* is the conjugate transpose of U, I is the identity matrix, and the multiplication is matrix multiplication.
Properties
Unitary matrices have several important properties that make them useful in various areas of mathematics and physics.
Norm Preservation
One of the most important properties of unitary matrices is that they preserve the norm of vectors. This means that if a vector is multiplied by a unitary matrix, the length (or norm) of the resulting vector is the same as the original vector.
Eigenvalues
The eigenvalues of a unitary matrix are complex numbers of absolute value 1. This follows directly from the definition of a unitary matrix and the properties of eigenvalues.
Determinant
The determinant of a unitary matrix is a complex number of absolute value 1. This can be proven using the properties of determinants and the definition of a unitary matrix.
Inverse
The inverse of a unitary matrix is its conjugate transpose. This follows directly from the definition of a unitary matrix.
Applications
Unitary matrices are used in a variety of fields, including physics, computer science, and engineering.
Quantum Mechanics
In quantum mechanics, unitary matrices are used to represent quantum states and transformations. This is because unitary matrices preserve the norm of vectors, which corresponds to the conservation of probability in quantum mechanics.
Signal Processing
In signal processing, unitary matrices are used in the design of filters and other signal processing algorithms. This is due to their norm-preserving property, which ensures that the energy of the signal is preserved.
Linear Algebra
In linear algebra, unitary matrices are used in various algorithms for matrix computations, such as the QR decomposition.