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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 .
Orthogonal matrix. In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is where QT is the transpose of Q and I is the identity matrix. This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to ...
hide. In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matrix being factorized is a normal or real symmetric matrix, the decomposition is called ...
Matrix decomposition. In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems.
Unitary transformation (quantum mechanics) In quantum mechanics, the Schrödinger equation describes how a system changes with time. It does this by relating changes in the state of the system to the energy in the system (given by an operator called the Hamiltonian). Therefore, once the Hamiltonian is known, the time dynamics are in principle ...
QR decomposition. In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix A into a product A = QR of an orthonormal matrix Q and an upper triangular matrix R. QR decomposition is often used to solve the linear least squares (LLS) problem and is the basis for a particular ...
Bottom: The action of Σ, a scaling by the singular values σ1 horizontally and σ2 vertically. Right: The action of U, another rotation. In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix into a rotation, followed by a rescaling followed by another rotation.
Also an orthogonal (real unitary) matrix has eigenvalues on the unit circle in the complex plane. Moreover, since its characteristic equation (an m th order polynomial in λ ) has real coefficients, it follows that its roots appear in complex conjugate pairs, that is, if α is a root then so is α ∗ .