Regular Matrix (other)
Regular matrix may refer to: Mathematics * Regular stochastic matrix, a stochastic matrix such that all the entries of some power of the matrix are positive * The opposite of irregular matrix, a matrix with a different number of entries in each row * Regular Hadamard matrix, a Hadamard matrix whose row and column sums are all equal * A regular element of a Lie algebra, when the Lie algebra is ''gln'' * Invertible matrix (this usage is rare) Other uses * QS Regular Matrix Quadraphonic Sound (originally called Quadphonic Synthesizer, and later referred to as RM or Regular Matrix) was a matrix 4-channel quadraphonic sound system for phonograph records. The system was based on technology created by Peter Scheiber, but ...

Stochastic Matrix
In mathematics, a stochastic matrix is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability. It is also called a probability matrix, transition matrix, substitution matrix, or Markov matrix. The stochastic matrix was first developed by Andrey Markov at the beginning of the 20th century, and has found use throughout a wide variety of scientific fields, including probability theory, statistics, mathematical finance and linear algebra, as well as computer science and population genetics. There are several different definitions and types of stochastic matrices: :A right stochastic matrix is a real square matrix, with each row summing to 1. :A left stochastic matrix is a real square matrix, with each column summing to 1. :A doubly stochastic matrix is a square matrix of nonnegative real numbers with each row and column summing to 1. In the same vein, one may define a stochastic vector (also ...
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Irregular Matrix
An irregular matrix, or ragged matrix, is a matrix that has a different number of elements in each row. Ragged matrices are not used in linear algebra, since standard matrix transformations cannot be performed on them, but they are useful as arrays in computing. Irregular matrices are typically stored using Iliffe vectors. For example, the following is an irregular matrix: : \begin 1 & 31 & 12& -3 \\ 7 & 2 \\ 1 & 2 & 2 \end See also * Regular matrix (other) * Empty matrix * Sparse matrix * Jagged array References * Paul E. BlackRagged matrix from Dictionary of Algorithms and Data Structures The NIST ''Dictionary of Algorithms and Data Structures'' is a reference work maintained by the U.S. National Institute of Standards and Technology. It defines a large number of terms relating to algorithms and data structures. For algorithms and ..., Paul E. Black, ed., NIST, 2004. {{Matrix classes Arrays Matrices ...
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Regular Hadamard Matrix
In mathematics a regular Hadamard matrix is a Hadamard matrix whose row and column sums are all equal. While the order of a Hadamard matrix must be 1, 2, or a multiple of 4, regular Hadamard matrices carry the further restriction that the order be a square number. The excess, denoted ''E''(''H''), of a Hadamard matrix ''H'' of order ''n'' is defined to be the sum of the entries of ''H''. The excess satisfies the bound , ''E''(''H''),  ≤ ''n''3/2. A Hadamard matrix attains this bound if and only if it is regular. Parameters If ''n'' = 4''u''2 is the order of a regular Hadamard matrix, then the excess is ±8''u''3 and the row and column sums all equal ±2''u''. It follows that each row has 2''u''2 ± ''u'' positive entries and 2''u''2 ∓ ''u'' negative entries. The orthogonality of rows implies that any two distinct rows have exactly ''u''2 ± ''u'' positive entries in common. If ''H'' is interpreted as the incidence matr ...
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Regular Element Of A Lie Algebra
In mathematics, a regular element of a Lie algebra or Lie group is an element whose centralizer has dimension as small as possible. For example, in a complex semisimple Lie algebra, an element X \in \mathfrak is regular if its centralizer in \mathfrak has dimension equal to the rank of \mathfrak, which in turn equals the dimension of some Cartan subalgebra \mathfrak (note that in earlier papers, an element of a complex semisimple Lie algebra was termed regular if it is semisimple and the kernel of its adjoint representation is a Cartan subalgebra). An element g \in G a Lie group is regular if its centralizer has dimension equal to the rank of G . Basic case In the specific case of \mathfrak_n(\mathbb), the Lie algebra of n \times n matrices over an algebraically closed field \mathbb(such as the complex numbers), a regular element M is an element whose Jordan normal form contains a single Jordan block for each eigenvalue (in other words, the geometric multiplicity of each eige ...
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Invertible Matrix
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix is uniquely determined by , and is called the (multiplicative) ''inverse'' of , denoted by . Matrix inversion is the process of finding the matrix that satisfies the prior equation for a given invertible matrix . A square matrix that is ''not'' invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is zero. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any finite region on the number line or complex plane, the probability that the matrix is singular is 0, that is, it will "almost never" be singular. Non-square matrices (-by- matrices for which ) do not hav ...
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