Multilinear spectral theory
... For matrices (order-2), only one way to take transpose (ie. swapping row and column indices) since Σ2 has only one non-trivial element. For an order-k tensor, there are k! − 1 different ‘transposes’ — one for each non-trivial element of Σk . An order-k tensor A = Jai1···ik K ∈ Rn×···×n is called sym ...
... For matrices (order-2), only one way to take transpose (ie. swapping row and column indices) since Σ2 has only one non-trivial element. For an order-k tensor, there are k! − 1 different ‘transposes’ — one for each non-trivial element of Σk . An order-k tensor A = Jai1···ik K ∈ Rn×···×n is called sym ...
570 SOME PROPERTIES OF THE DISCRIMINANT MATRICES OF A
... where ti(eres) and fa{erea) are the first and second traces, respectively, of eres. The first forms in terms of the constants of multiplication arise from the isomorphism between the first and second matrices of the elements of A and the elements themselves. The second forms result from direct calcu ...
... where ti(eres) and fa{erea) are the first and second traces, respectively, of eres. The first forms in terms of the constants of multiplication arise from the isomorphism between the first and second matrices of the elements of A and the elements themselves. The second forms result from direct calcu ...
Lecture 2. Solving Linear Systems
... the system admits a unique solution if and only if there is no free variable. Theorem 2.4. Consider a linear system whose augmented matrix is M: There are three possibilities. (1) If the last column of M is a pivot column, then the system is inconsistent. (2) If the last column of M is a not pivot c ...
... the system admits a unique solution if and only if there is no free variable. Theorem 2.4. Consider a linear system whose augmented matrix is M: There are three possibilities. (1) If the last column of M is a pivot column, then the system is inconsistent. (2) If the last column of M is a not pivot c ...
A Colorful Introduction to Linear Algebra - Mine
... When we teach children to count we might say: “one ball, two balls, three balls”. Then we might say: “one block, two blocks, three blocks”. The child might reply: “one ball, two blocks, three dolls” and we correct them: “No, no you have to add things of the same type!” It is meaningless to a ...
... When we teach children to count we might say: “one ball, two balls, three balls”. Then we might say: “one block, two blocks, three blocks”. The child might reply: “one ball, two blocks, three dolls” and we correct them: “No, no you have to add things of the same type!” It is meaningless to a ...
Fixed Point
... Together, these results imply that there exists a subordinate matrix norm k·k on Rn×n such that kAk < 1, if and only if ρ(A) < 1. Thus, the main condition of the previous theorem could also be equivalently reformulated as ...
... Together, these results imply that there exists a subordinate matrix norm k·k on Rn×n such that kAk < 1, if and only if ρ(A) < 1. Thus, the main condition of the previous theorem could also be equivalently reformulated as ...
MATH3033 Information Sheet 1. The Standard Maximum Problem
... the tied entries) and pivot on ai0 j0 . 9. Sensitivity Analysis Suppose the standard maximum problem has been solved with dual solution y = (y1 , . . . , ym ). To calculate the new CPF solution under changes in parameter(s) A → Â, b → b̂ and c → ĉ, follow the following steps: (a) Construct the mat ...
... the tied entries) and pivot on ai0 j0 . 9. Sensitivity Analysis Suppose the standard maximum problem has been solved with dual solution y = (y1 , . . . , ym ). To calculate the new CPF solution under changes in parameter(s) A → Â, b → b̂ and c → ĉ, follow the following steps: (a) Construct the mat ...
Approximation Algorithms and Semidefinite Programming
... or π. Moreover, the model assumes that arithmetic operations on real numbers (including computations of square roots, trigonometric functions, etc.) take constant time. The model is motivated by actual computers that approximate the real numbers by floating-point numbers with fixed precision. The re ...
... or π. Moreover, the model assumes that arithmetic operations on real numbers (including computations of square roots, trigonometric functions, etc.) take constant time. The model is motivated by actual computers that approximate the real numbers by floating-point numbers with fixed precision. The re ...
CHARACTERISTIC ROOTS AND FIELD OF VALUES OF A MATRIX
... Beginning with Bendixson [3] in 1900, many writers have obtained limits for the characteristic roots of a matrix. In many cases these were actually limits for the field of values of the matrix [14]. In an address delivered before the Mathematical Association of America in 1938, Browne [10] gave a su ...
... Beginning with Bendixson [3] in 1900, many writers have obtained limits for the characteristic roots of a matrix. In many cases these were actually limits for the field of values of the matrix [14]. In an address delivered before the Mathematical Association of America in 1938, Browne [10] gave a su ...
Package `sparseHessianFD`
... logical matrices, and FALSE for pattern matrices. If M is a pattern matrix, values=TRUE will trigger a warning. ...
... logical matrices, and FALSE for pattern matrices. If M is a pattern matrix, values=TRUE will trigger a warning. ...
A fast algorithm for approximate polynomial gcd based on structured
... they are prone to numerical instabilities; look-ahead strategies can improve the numerical stability with an increase of the complexity to O(n3 ). More recently, O(n2 ) algorithms have been proposed in [24] and [16]. They are based on the QR factorization of a displacement structured matrix obtained ...
... they are prone to numerical instabilities; look-ahead strategies can improve the numerical stability with an increase of the complexity to O(n3 ). More recently, O(n2 ) algorithms have been proposed in [24] and [16]. They are based on the QR factorization of a displacement structured matrix obtained ...
Vectors and Matrices
... Definition. Two matrices A and B are said to be equal, written A = B, if they have the same dimension and their corresponding elements are equal, i.e., ai j = bi j for all i and j. In some instances it is convenient to think of vectors as merely being special cases of matrices. However, we will late ...
... Definition. Two matrices A and B are said to be equal, written A = B, if they have the same dimension and their corresponding elements are equal, i.e., ai j = bi j for all i and j. In some instances it is convenient to think of vectors as merely being special cases of matrices. However, we will late ...
SECOND-ORDER VERSUS FOURTH
... the array output covariance matrix and C in place of the signal covariance. Since only 4th-order cumulants are used, the properties of the measurement noise do not appear in the expression of the exact statistics. Q (M) is rank defective, with a rank equal to the rank of C, and its range is spanned ...
... the array output covariance matrix and C in place of the signal covariance. Since only 4th-order cumulants are used, the properties of the measurement noise do not appear in the expression of the exact statistics. Q (M) is rank defective, with a rank equal to the rank of C, and its range is spanned ...
Non-negative matrix factorization
NMF redirects here. For the bridge convention, see new minor forcing.Non-negative matrix factorization (NMF), also non-negative matrix approximation is a group of algorithms in multivariate analysis and linear algebra where a matrix V is factorized into (usually) two matrices W and H, with the property that all three matrices have no negative elements. This non-negativity makes the resulting matrices easier to inspect. Also, in applications such as processing of audio spectrograms non-negativity is inherent to the data being considered. Since the problem is not exactly solvable in general, it is commonly approximated numerically.NMF finds applications in such fields as computer vision, document clustering, chemometrics, audio signal processing and recommender systems.