Details and Options
- A p×q matrix m is unitary if p≥q and ConjugateTranspose[m].m is the q×q identity matrix, or p≤q and m.ConjugateTranspose[m] is the p×p identity matrix.
- UnitaryMatrixQ works for symbolic as well as numerical matrices.
- The following options can be given:
Normalized True test if matrix columns are normalized SameTest Automatic function to test equality of expressions Tolerance Automatic tolerance for approximate numbers
- For exact and symbolic matrices, the option SameTest->f indicates that two entries aij and bij are taken to be equal if f[aij,bij] gives True.
- For approximate matrices, the option Tolerance->t can be used to indicate that the norm γ=m.m-In∞ satisfying γ≤t is taken to be zero where In is the identity matrix.
Examplesopen allclose all
Basic Examples (2)
Basic Uses (6)
Use UnitaryMatrixQ with an arbitrary-precision matrix:
Use UnitaryMatrixQ with a symbolic matrix:
UnitaryMatrixQ works efficiently with large numerical matrices:
Special Matrices (4)
Rectangular Semi-unitary Matrices (4)
However, it will not give true for ConjugateTranspose[m]:
Sources of Unitary Matrices (4)
Orthogonalize applied to linearly independent, complex vectors generates a unitary matrix:
Matrices drawn from CircularUnitaryMatrixDistribution are unitary:
So are matrices drawn from CircularOrthogonalMatrixDistribution:
As are matrices from CircularSymplecticMatrixDistribution:
Uses of Orthogonal Matrices (5)
In quantum mechanics, time evolution is represented by a 1-parameter family of unitary matrices . times the logarithmic derivative of is a Hermitian matrix called the Hamiltonian or energy operator . Its eigenvalues represent the possible energies of the system. For the following time evolution, compute the Hamiltonian and possible energies:
The exponential MatrixExp[v] of an antihermitian matrix is unitary. Define a matrix function through its differential equation with initial value and show that the solution is unitary:
Properties & Relations (13)
Use Eigenvalues to find eigenvalues:
Use Eigenvectors to find eigenvectors:
Wolfram Research (2014), UnitaryMatrixQ, Wolfram Language function, https://reference.wolfram.com/language/ref/UnitaryMatrixQ.html.
Wolfram Language. 2014. "UnitaryMatrixQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/UnitaryMatrixQ.html.
Wolfram Language. (2014). UnitaryMatrixQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/UnitaryMatrixQ.html