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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 = =, 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
The GNM normal modes are found by diagonalization of the Kirchhoff matrix, Γ = UΛU T. Here, U is a unitary matrix, U T = U −1, of the eigenvectors u i of Γ and Λ is the diagonal matrix of eigenvalues λ i. The frequency and shape of a mode is represented by its eigenvalue and eigenvector, respectively.
The non-chiral Su–Schrieffer–Heeger model (=), can be associated with symmetry class BDI with an integer topological invariant due to gauge invariance. [6] [7] The problem is similar to the integer quantum Hall effect and the quantum anomalous Hall effect (both in =) which are A class, with integer Chern number.
Two complex Hadamard matrices are called equivalent, written , if there exist diagonal unitary matrices , and permutation matrices, such that =. Any complex Hadamard matrix is equivalent to a dephased Hadamard matrix, in which all elements in the first row and first column are equal to unity.
To this end, we will construct a 20x20 matrix where the (,) th entry is equal to the probability of the th amino acid being transformed into the th amino acid in a certain amount of evolutionary time. There are many different ways to construct such a matrix, called a substitution matrix. Here are the most commonly used ones:
More generally, unitary matrices are precisely the unitary operators on finite-dimensional Hilbert spaces, so the notion of a unitary operator is a generalization of the notion of a unitary matrix. Orthogonal matrices are the special case of unitary matrices in which all entries are real. [4] They are the unitary operators on R n.
In biology, matrix (pl.: matrices) is the material (or tissue) in between a eukaryotic organism's cells. The structure of connective tissues is an extracellular matrix. Fingernails and toenails grow from matrices. It is found in various connective tissues. It serves as a jelly-like structure instead of cytoplasm in connective tissue.
Specifically, the singular value decomposition of an complex matrix is a factorization of the form =, where is an complex unitary matrix, is an rectangular diagonal matrix with non-negative real numbers on the diagonal, is an complex unitary matrix, and is the conjugate transpose of . Such decomposition ...