pMatlab v2.0 Function Reference Table of Contents

... Government. MATLAB ® is a registered trademark of The Mathworks, Inc. ...

... Government. MATLAB ® is a registered trademark of The Mathworks, Inc. ...

Sheldon Axler Third Edition

... deﬁned in this chapter as the sum of the eigenvalues and the product of the eigenvalues, both counting multiplicity. These easy-to-remember deﬁnitions would not be possible with the traditional approach to eigenvalues, because the traditional method uses determinants to prove that sufﬁcient eigenval ...

... deﬁned in this chapter as the sum of the eigenvalues and the product of the eigenvalues, both counting multiplicity. These easy-to-remember deﬁnitions would not be possible with the traditional approach to eigenvalues, because the traditional method uses determinants to prove that sufﬁcient eigenval ...

Symmetric tensors and symmetric tensor rank

... tensors of symmetric rank not more than r is not closed, unless r = 1 (see Sections 6 and 8) — a fact that parallels the illposedness discussed in [21]. The focus of this paper is mainly on symmetric tensors. The asymmetric case will be addressed in a companion paper, and will use similar tools borr ...

... tensors of symmetric rank not more than r is not closed, unless r = 1 (see Sections 6 and 8) — a fact that parallels the illposedness discussed in [21]. The focus of this paper is mainly on symmetric tensors. The asymmetric case will be addressed in a companion paper, and will use similar tools borr ...

Regularization of Least Squares Problems

... scaled such that σj ≥ 0 are nonnegative and are ordered by magnitude: σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0. σi , i = 1, . . . , n are the singular values of A, the columns of U are the left singular vectors and the columns of V are the right singular vectors of A. TUHH ...

... scaled such that σj ≥ 0 are nonnegative and are ordered by magnitude: σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0. σi , i = 1, . . . , n are the singular values of A, the columns of U are the left singular vectors and the columns of V are the right singular vectors of A. TUHH ...

SEMIDEFINITE DESCRIPTIONS OF THE CONVEX HULL OF

... this paper is the polar of SO(n), the set of linear functionals that take value at most one on SO(n), i.e., SO(n)◦ = {Y ∈ Rn×n : Y, X ≤ 1 for all X ∈ SO(n)}, where we have identiﬁed Rn×n with its dual space via the trace inner product Y, X = tr(Y T X). These two convex bodies are closely related ...

... this paper is the polar of SO(n), the set of linear functionals that take value at most one on SO(n), i.e., SO(n)◦ = {Y ∈ Rn×n : Y, X ≤ 1 for all X ∈ SO(n)}, where we have identiﬁed Rn×n with its dual space via the trace inner product Y, X = tr(Y T X). These two convex bodies are closely related ...

NP-hardness of Deciding Convexity of Quartic Polynomials and

... Perhaps the simplest example is in global minimization of polynomials, where it could be very useful to decide first whether the polynomial to be optimized is convex. Once convexity is verified, then every local minimum is global and very basic techniques (e.g., gradient descent) can find a global m ...

... Perhaps the simplest example is in global minimization of polynomials, where it could be very useful to decide first whether the polynomial to be optimized is convex. Once convexity is verified, then every local minimum is global and very basic techniques (e.g., gradient descent) can find a global m ...

Writing MATLAB C/MEX Code

... The output variables are initially unassigned; it is the responsibility of the MEX-function to create them. So for the example [X,Y] = mymexfun(A,B,C) it is our responsibility to create X and Y. If nlhs = 0, the MEX-function is still allowed return one output variable, in which case plhs[0] represen ...

... The output variables are initially unassigned; it is the responsibility of the MEX-function to create them. So for the example [X,Y] = mymexfun(A,B,C) it is our responsibility to create X and Y. If nlhs = 0, the MEX-function is still allowed return one output variable, in which case plhs[0] represen ...

An exponential-related function for decision-making in

... agement of vagueness and uncertainty. As indicated by Wan and Li [2] and Aikhuele & Turan [3], the IFS are more adaptable, functional and capable than the traditional FS theory at handling uncertainty and vagueness in practices. The advantages of applying the IFS have been reported in [5] to include ...

... agement of vagueness and uncertainty. As indicated by Wan and Li [2] and Aikhuele & Turan [3], the IFS are more adaptable, functional and capable than the traditional FS theory at handling uncertainty and vagueness in practices. The advantages of applying the IFS have been reported in [5] to include ...

MODULAR ARITHMETIC 1. Introduction

... Example 2.5. Taking m = 2, every integer is congruent modulo 2 to exactly one of 0 and 1. Saying n ≡ 0 mod 2 means n = 2k for some integer k, so n is even, and saying n ≡ 1 mod 2 means n = 2k + 1 for some integer k, so n is odd. We have a ≡ b mod 2 precisely when a and b have the same parity: both a ...

... Example 2.5. Taking m = 2, every integer is congruent modulo 2 to exactly one of 0 and 1. Saying n ≡ 0 mod 2 means n = 2k for some integer k, so n is even, and saying n ≡ 1 mod 2 means n = 2k + 1 for some integer k, so n is odd. We have a ≡ b mod 2 precisely when a and b have the same parity: both a ...

(pdf)

... ⇒ (T − λ j I) k+1 (T − λ j I) n x = 0 ⇒ (T − λ j I) n x ∈ null(T − λ j I) k+1 = (T − λ j I) k . Thus null(T − λ j I) k+n+1 ⊆ null(T − λ j I)k+n ⇒ null(T − λ j I) k+n+1 = null(T − λ j I)k+n . So null(T − λ j I) k = null(T − λ j I) k+1 ⇒ null(T − λ j I)1 ⊆ null(T − λ j I) 2 ⊆ … null(T − λ j I) k = nul ...

... ⇒ (T − λ j I) k+1 (T − λ j I) n x = 0 ⇒ (T − λ j I) n x ∈ null(T − λ j I) k+1 = (T − λ j I) k . Thus null(T − λ j I) k+n+1 ⊆ null(T − λ j I)k+n ⇒ null(T − λ j I) k+n+1 = null(T − λ j I)k+n . So null(T − λ j I) k = null(T − λ j I) k+1 ⇒ null(T − λ j I)1 ⊆ null(T − λ j I) 2 ⊆ … null(T − λ j I) k = nul ...

COMPUTATIONS FOR ALGEBRAS AND GROUP

... of bases for the simple components of a semi-simple algebra, and the expression of a simple algebra as a full matrix ring over a division algebra. Building on the work of Friedl and Rónyai, we present a simple and efficient probabilistic algorithm for the computation of simple components of a semi- ...

... of bases for the simple components of a semi-simple algebra, and the expression of a simple algebra as a full matrix ring over a division algebra. Building on the work of Friedl and Rónyai, we present a simple and efficient probabilistic algorithm for the computation of simple components of a semi- ...

# 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.