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Polar Decomposition Theorem
In mathematics, the polar decomposition of a square real or complex matrix A is a factorization of the form A = U P, where U is an orthogonal matrix and P is a positive semi-definite symmetric matrix (U is a unitary matrix and P is a positive semi-definite Hermitian matrix in the complex case), both square and of the same size. Intuitively, if a real n\times n matrix A is interpreted as a linear transformation of n-dimensional space \mathbb^n, the polar decomposition separates it into a rotation or reflection U of \mathbb^n, and a scaling of the space along a set of n orthogonal axes. The polar decomposition of a square matrix A always exists. If A is invertible, the decomposition is unique, and the factor P will be positive-definite. In that case, A can be written uniquely in the form A = U e^X , where U is unitary and X is the unique self-adjoint logarithm of the matrix P. This decomposition is useful in computing the fundamental group of (matrix) Lie groups. The polar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have isomorphic fundamental groups. The fundamental group of a topological space X is denoted by \pi_1(X). Intuition Start with a space (for example, a surface), and some point in it, and all the loops both starting and ending at this point— paths that start at this point, wander around and eventually return to the starting point. Two loops can be combined in an obvious way: travel along the first loop, then along the second. Two loops are considered equivalent if one can be deformed into the other without breakin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commuting Matrices
In linear algebra, two matrices A and B are said to commute if AB=BA, or equivalently if their commutator ,B AB-BA is zero. A set of matrices A_1, \ldots, A_k is said to commute if they commute pairwise, meaning that every pair of matrices in the set commute with each other. Characterizations and properties * Commuting matrices preserve each other's eigenspaces. As a consequence, commuting matrices over an algebraically closed field are simultaneously triangularizable; that is, there are bases over which they are both upper triangular. In other words, if A_1,\ldots,A_k commute, there exists a similarity matrix P such that P^ A_i P is upper triangular for all i \in \. The converse is not necessarily true, as the following counterexample shows: *:\begin 1 & 2 \\ 0 & 3 \end\begin 1 & 1 \\ 0 & 1 \end = \begin 1 & 3 \\ 0 & 3 \end \ne \begin 1 & 5 \\ 0 & 3 \end=\begin 1 & 1 \\ 0 & 1 \end\begin 1 & 2 \\ 0 & 3 \end. : However, if the square of the commutator of two matrices is zero, that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Matrix
In mathematics, a complex square matrix is normal if it commutes with its conjugate transpose : The concept of normal matrices can be extended to normal operators on infinite dimensional normed spaces and to normal elements in C*-algebras. As in the matrix case, normality means commutativity is preserved, to the extent possible, in the noncommutative setting. This makes normal operators, and normal elements of C*-algebras, more amenable to analysis. The spectral theorem states that a matrix is normal if and only if it is unitarily similar to a diagonal matrix, and therefore any matrix satisfying the equation is diagonalizable. The converse does not hold because diagonalizable matrices may have non-orthogonal eigenspaces. The left and right singular vectors in the singular value decomposition of a normal matrix \mathbf = \mathbf \boldsymbol \mathbf^* differ only in complex phase from each other and from the corresponding eigenvectors, since the phase must be factored out ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive-semidefinite Matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number z^* Mz is positive for every nonzero complex column vector z, where z^* denotes the conjugate transpose of z. Positive semi-definite matrices are defined similarly, except that the scalars z^\textsfMz and z^* Mz are required to be positive ''or zero'' (that is, nonnegative). Negative-definite and negative semi-definite matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called indefinite. A matrix is thus positive-definite if and only if it is the matrix of a positive-definite quadratic form or Hermitian form. In other words, a matrix is positive-definite if and only if it defines a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Value Decomposition
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is related to the polar decomposition. Specifically, the singular value decomposition of an \ m \times n\ complex matrix is a factorization of the form \ \mathbf = \mathbf\ , where is an \ m \times m\ complex unitary matrix, \ \mathbf\ is an \ m \times n\ rectangular diagonal matrix with non-negative real numbers on the diagonal, is an n \times n complex unitary matrix, and \ \mathbf\ is the conjugate transpose of . Such decomposition always exists for any complex matrix. If is real, then and can be guaranteed to be real orthogonal matrices; in such contexts, the SVD is often denoted \ \mathbf^\mathsf\ . The diagonal entries \ \sigma_i = \Sigma_\ of \ \mathbf\ are uniquely determined by and are known as the singular values of . The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Root Of A Matrix
In mathematics, the square root of a matrix extends the notion of square root from numbers to matrices. A matrix is said to be a square root of if the matrix product is equal to . Some authors use the name ''square root'' or the notation only for the specific case when is positive semidefinite, to denote the unique matrix that is positive semidefinite and such that (for real-valued matrices, where is the transpose of ). Less frequently, the name ''square root'' may be used for any factorization of a positive semidefinite matrix as , as in the Cholesky factorization, even if . This distinct meaning is discussed in '. Examples In general, a matrix can have several square roots. In particular, if A = B^2 then A=(-B)^2 as well. The 2×2 identity matrix \textstyle\begin1 & 0\\ 0 & 1\end has infinitely many square roots. They are given by :\begin \pm 1 & 0\\ 0 & \pm 1\end and \begin a & b\\ c & -a\end where (a, b, c) are any numbers (real or complex) such that a^2+bc= ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate Transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex conjugate of a+ib being a-ib, for real numbers a and b). It is often denoted as \boldsymbol^\mathrm or \boldsymbol^* or \boldsymbol'. H. W. Turnbull, A. C. Aitken, "An Introduction to the Theory of Canonical Matrices," 1932. For real matrices, the conjugate transpose is just the transpose, \boldsymbol^\mathrm = \boldsymbol^\mathsf. Definition The conjugate transpose of an m \times n matrix \boldsymbol is formally defined by where the subscript ij denotes the (i,j)-th entry, for 1 \le i \le n and 1 \le j \le m, and the overbar denotes a scalar complex conjugate. This definition can also be written as :\boldsymbol^\mathrm = \left(\overline\right)^\mathsf = \overline where \boldsymbol^\mathsf denotes the transpose and \overline denotes the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Matrices
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the linear map represented by the matrix is an isomorphism. The determinant of a product of matrices is the product of their determinants (the preceding property is a corollary of this one). The determinant of a matrix is denoted , , or . The determinant of a matrix is :\begin a & b\\c & d \end=ad-bc, and the determinant of a matrix is : \begin a & b & c \\ d & e & f \\ g & h & i \end= aei + bfg + cdh - ceg - bdi - afh. The determinant of a matrix can be defined in several equivalent ways. Leibniz formula expresses the determinant as a sum of signed products of matrix entries such that each summand is the product of different entries, and the number of these summands is n!, the factorial of (t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - bi. The complex conjugate of z is often denoted as \overline or z^*. In polar form, the conjugate of r e^ is r e^. This can be shown using Euler's formula. The product of a complex number and its conjugate is a real number: a^2 + b^2 (or r^2 in polar coordinates). If a root of a univariate polynomial with real coefficients is complex, then its complex conjugate is also a root. Notation The complex conjugate of a complex number z is written as \overline z or z^*. The first notation, a vinculum, avoids confusion with the notation for the conjugate transpose of a matrix, which can be thought of as a generalization of the complex conjugate. The second is preferred in physics, where dagger (†) is used for the conjugate tra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semi-orthogonal Matrix
In linear algebra, a semi-orthogonal matrix is a non-square matrix with real entries where: if the number of columns exceeds the number of rows, then the rows are orthonormal vectors; but if the number of rows exceeds the number of columns, then the columns are orthonormal vectors. Equivalently, a non-square matrix ''A'' is semi-orthogonal if either :A^ A = I \text A A^ = I. \,Povey, Daniel, et al. (2018)"Semi-Orthogonal Low-Rank Matrix Factorization for Deep Neural Networks."Interspeech. In the following, consider the case where ''A'' is an ''m'' × ''n'' matrix for ''m'' > ''n''. Then :A^ A = I_n, \text :A A^ = \text A. The fact that A^ A = I_n implies the isometry property :\, A x\, _2 = \, x\, _2 \, for all ''x'' in R''n''. For example, \begin1 \\ 0\end is a semi-orthogonal matrix. A semi-orthogonal matrix ''A'' is semi-unitary (either ''A''†''A'' = ''I'' or ''AA''† = ''I'') and either left-invertible or right-invertible (l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
