Symmetric matrixIn linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if denotes the entry in the th row and th column then for all indices and Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative.
Skew-symmetric matrixIn mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition In terms of the entries of the matrix, if denotes the entry in the -th row and -th column, then the skew-symmetric condition is equivalent to The matrix is skew-symmetric because Throughout, we assume that all matrix entries belong to a field whose characteristic is not equal to 2.
Orthogonal matrixIn 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 its inverse: where Q−1 is the inverse of Q. An orthogonal matrix Q is necessarily invertible (with inverse Q−1 = QT), unitary (Q−1 = Q∗), where Q∗ is the Hermitian adjoint (conjugate transpose) of Q, and therefore normal (Q∗Q = QQ∗) over the real numbers.
EngineeringEngineering is the practice of using natural science, mathematics, and the engineering design process to solve problems, increase efficiency and productivity, and improve systems. Modern engineering comprises many subfields which include designing and creating infrastructure, machinery, vehicles, electronics, materials, and energy. The discipline of engineering encompasses a broad range of more specialized fields of engineering, each with a more specific emphasis on particular areas of applied mathematics, applied science, and types of application.
Pauli matricesIn mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (σ), they are occasionally denoted by tau (τ) when used in connection with isospin symmetries. These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equation which takes into account the interaction of the spin of a particle with an external electromagnetic field.
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