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Sylvester Equation
In mathematics, in the field of control theory, a Sylvester equation is a Matrix (mathematics), matrix equation of the form: :A X + X B = C. Then given matrices ''A'', ''B'', and ''C'', the problem is to find the possible matrices ''X'' that obey this equation. All matrices are assumed to have coefficients in the complex numbers. For the equation to make sense, the matrices must have appropriate sizes, for example they could all be square matrices of the same size. But more generally, ''A'' and ''B'' must be square matrices of sizes ''n'' and ''m'' respectively, and then ''X'' and ''C'' both have ''n'' rows and ''m'' columns. A Sylvester equation has a unique solution for ''X'' exactly when there are no common eigenvalues of ''A'' and −''B''. More generally, the equation ''AX'' + ''XB'' = ''C'' has been considered as an equation of bounded operators on a (possibly infinite-dimensional) Banach space. In this case, the condition for the uniqueness of a solut ... [...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] |
Cayley–Hamilton Theorem
In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) satisfies its own characteristic equation. If is a given matrix and is the identity matrix, then the characteristic polynomial of is defined as p_A(\lambda)=\det(\lambda I_n-A), where is the determinant operation and is a variable for a scalar element of the base ring. Since the entries of the matrix (\lambda I_n-A) are (linear or constant) polynomials in , the determinant is also a degree- monic polynomial in , p_A(\lambda) = \lambda^n + c_\lambda^ + \cdots + c_1\lambda + c_0~. One can create an analogous polynomial p_A(A) in the matrix instead of the scalar variable , defined as p_A(A) = A^n + c_A^ + \cdots + c_1A + c_0I_n~. The Cayley–Hamilton theorem states that this polynomial expression is equal to the zero matrix, which is to say tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bull
A bull is an intact (i.e., not castrated) adult male of the species ''Bos taurus'' (cattle). More muscular and aggressive than the females of the same species (i.e., cows), bulls have long been an important symbol in many religions, including for sacrifices. These animals play a significant role in beef ranching, dairy farming, and a variety of sporting and cultural activities, including bullfighting and bull riding. Due to their temperament, handling requires precautions. Nomenclature The female counterpart to a bull is a cow, while a male of the species that has been castrated is a ''steer'', '' ox'', or ''bullock'', although in North America, this last term refers to a young bull. Use of these terms varies considerably with area and dialect. Colloquially, people unfamiliar with cattle may refer to both castrated and intact animals as "bulls". A wild, young, unmarked bull is known as a ''micky'' in Australia.Sheena Coupe (ed.), ''Frontier Country, Vol. 1'' (Weldon R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Comm
The command in the Unix family of computer operating systems is a utility that is used to compare two files for common and distinct lines. is specified in the POSIX standard. It has been widely available on Unix-like operating systems since the mid to late 1980s. History Written by Lee E. McMahon, first appeared in Version 4 Unix. The version of bundled in GNU coreutils was written by Richard Stallman and David MacKenzie. Usage reads two files as input, regarded as lines of text. outputs one file, which contains three columns. The first two columns contain lines unique to the first and second file, respectively. The last column contains lines common to both. This functionally is similar to . Columns are typically distinguished with the character. If the input files contain lines beginning with the separator character, the output columns can become ambiguous. For efficiency, standard implementations of expect both input files to be sequenced in the same line collat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lyapunov Equation
In control theory, the discrete Lyapunov equation is of the form :A X A^ - X + Q = 0 where Q is a Hermitian matrix and A^H is the conjugate transpose of A. The continuous Lyapunov equation is of the form :AX + XA^H + Q = 0. The Lyapunov equation occurs in many branches of control theory, such as stability analysis and optimal control. This and related equations are named after the Russian mathematician Aleksandr Lyapunov. Application to stability In the following theorems A, P, Q \in \mathbb^, and P and Q are symmetric. The notation P>0 means that the matrix P is positive definite. Theorem (continuous time version). Given any Q>0, there exists a unique P>0 satisfying A^T P + P A + Q = 0 if and only if the linear system \dot=A x is globally asymptotically stable. The quadratic function V(x)=x^T P x is a Lyapunov function that can be used to verify stability. Theorem (discrete time version). Given any Q>0, there exists a unique P>0 satisfying A^T P A -P + Q = 0 if and only if t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
IEEE Transactions On Image Processing
The ''IEEE Transactions on Image Processing'' is a monthly peer-reviewed scientific journal covering aspects of image processing in the field of signal processing. It was established in 1992 and is published by the IEEE Signal Processing Society. The editor-in-chief is Alessandro Foi (Tampere University). According to the ''Journal Citation Reports'', the journal has a 2020 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as ... of 10.856. References External links * Image processing IEEE academic journals Monthly journals Publications established in 1992 Computer science journals English-language journals {{computer-science-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GNU Octave
GNU Octave is a high-level programming language primarily intended for scientific computing and numerical computation. Octave helps in solving linear and nonlinear problems numerically, and for performing other numerical experiments using a language that is mostly compatible with MATLAB. It may also be used as a batch-oriented language. As part of the GNU Project, it is free software under the terms of the GNU General Public License. History The project was conceived around 1988. At first it was intended to be a companion to a chemical reactor design course. Full development was started by John W. Eaton in 1992. The first alpha release dates back to 4 January 1993 and on 17 February 1994 version 1.0 was released. Version 7.1.0 was released on Apr 6, 2022. The program is named after Octave Levenspiel, a former professor of the principal author. Levenspiel was known for his ability to perform quick back-of-the-envelope calculations. Development history Developments In addition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LAPACK
LAPACK ("Linear Algebra Package") is a standard software library for numerical linear algebra. It provides routines for solving systems of linear equations and linear least squares, eigenvalue problems, and singular value decomposition. It also includes routines to implement the associated matrix factorizations such as LU, QR, Cholesky and Schur decomposition. LAPACK was originally written in FORTRAN 77, but moved to Fortran 90 in version 3.2 (2008). The routines handle both real and complex matrices in both single and double precision. LAPACK relies on an underlying BLAS implementation to provide efficient and portable computational building blocks for its routines. LAPACK was designed as the successor to the linear equations and linear least-squares routines of LINPACK and the eigenvalue routines of EISPACK. LINPACK, written in the 1970s and 1980s, was designed to run on the then-modern vector computers with shared memory. LAPACK, in contrast, was designed to effectivel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big O Notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for ''Ordnung'', meaning the order of approximation. In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetical function and a better understood approximation; a famous example of such a difference is the remainder term in the prime number theorem. Big O notation is also used in many other fields to provide similar estimates. Big O notation characterizes functions according to their growth rates: d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Matrix
In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal are zero. Because matrix equations with triangular matrices are easier to solve, they are very important in numerical analysis. By the LU decomposition algorithm, an invertible matrix may be written as the product of a lower triangular matrix ''L'' and an upper triangular matrix ''U'' if and only if all its leading principal minors are non-zero. Description A matrix of the form :L = \begin \ell_ & & & & 0 \\ \ell_ & \ell_ & & & \\ \ell_ & \ell_ & \ddots & & \\ \vdots & \vdots & \ddots & \ddots & \\ \ell_ & \ell_ & \ldots & \ell_ & \ell_ \end is called a lower triangular matrix or left triangular matrix, and a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
QR Algorithm
In numerical linear algebra, the QR algorithm or QR iteration is an eigenvalue algorithm: that is, a procedure to calculate the eigenvalues and eigenvectors of a matrix. The QR algorithm was developed in the late 1950s by John G. F. Francis and by Vera N. Kublanovskaya, working independently. The basic idea is to perform a QR decomposition, writing the matrix as a product of an orthogonal matrix and an upper triangular matrix, multiply the factors in the reverse order, and iterate. The practical QR algorithm Formally, let ''A'' be a real matrix of which we want to compute the eigenvalues, and let ''A''0:=''A''. At the ''k''-th step (starting with ''k'' = 0), we compute the QR decomposition ''A''''k''=''Q''''k''''R''''k'' where ''Q''''k'' is an orthogonal matrix (i.e., ''Q''''T'' = ''Q''−1) and ''R''''k'' is an upper triangular matrix. We then form ''A''''k''+1 = ''R''''k''''Q''''k''. Note that : A_ = R_k Q_k = Q_k^ Q_k R_k Q_k = Q_k^ A_k Q_k = Q_k^ A_k Q_k, so all the ''A''''k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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