Exam Review 1 Spring 16, 21-241: Matrices and Linear Transformations
... 14. Let P be the powerset of N, the set of natural numbers. Consider the vector space {P, F, 4, } where F consists of {0, 1} with addition defined modulo 2, 4 is the symmetric set difference, and is such that 1 S = S and 0 S is the zero vector. (a) Let S ∈ P . What must 1 S be? (b) What is ...
... 14. Let P be the powerset of N, the set of natural numbers. Consider the vector space {P, F, 4, } where F consists of {0, 1} with addition defined modulo 2, 4 is the symmetric set difference, and is such that 1 S = S and 0 S is the zero vector. (a) Let S ∈ P . What must 1 S be? (b) What is ...
Solving Systems of Equations
... Matrices and Systems of Equations In solving these linear systems, you’ll note that we manipulate the coefficients, not necessarily the variables. The variables just hold the place, so to speak. A matrix is an array of numbers. We define the size (dimension) of a matrix to be the number of rows by t ...
... Matrices and Systems of Equations In solving these linear systems, you’ll note that we manipulate the coefficients, not necessarily the variables. The variables just hold the place, so to speak. A matrix is an array of numbers. We define the size (dimension) of a matrix to be the number of rows by t ...
Matrix Multiplication
... order. Thus, to calculate, say, ABC, you can first form AB and then multiply this result from the right by matrix C, or, you can first form BC and then multiply this result by A from the left. The final result will be the same (see exercise 4 below). iii) The square matrix with diagonal elements ...
... order. Thus, to calculate, say, ABC, you can first form AB and then multiply this result from the right by matrix C, or, you can first form BC and then multiply this result by A from the left. The final result will be the same (see exercise 4 below). iii) The square matrix with diagonal elements ...
Back matter - Ohio University Department of Mathematics
... Plots level curves of a function of two variables. Filled contour plot. Easy contour plot. Creates a log-log plot. Draws a mesh surface. Creates arrays that can be used as inputs in graphics commands such as contour, mesh, quiver, and surf. Easy mesh surface plot. Plots data vectors. Easy plot for s ...
... Plots level curves of a function of two variables. Filled contour plot. Easy contour plot. Creates a log-log plot. Draws a mesh surface. Creates arrays that can be used as inputs in graphics commands such as contour, mesh, quiver, and surf. Easy mesh surface plot. Plots data vectors. Easy plot for s ...
Notes
... If A is diagonal, then the terms in the diagonal will be the solution to the characteristic eq. The determinant is a polynomial of degree n and has i as the latent roots. The product of i equals the determinant of A, and the sum i equals the trace of A. All vectors are called characteristic (eige ...
... If A is diagonal, then the terms in the diagonal will be the solution to the characteristic eq. The determinant is a polynomial of degree n and has i as the latent roots. The product of i equals the determinant of A, and the sum i equals the trace of A. All vectors are called characteristic (eige ...
(Slide 1) Question 10
... (Slide 1) Question 10. What is the semi-mechanistic model? Answer In the last few years the semi-mechanistic model which is based on key embrittlement mechanisms of chemical elements influence is proposed. The model includes three basic mechanisms influencing to radiation embrittlement for RPV steel ...
... (Slide 1) Question 10. What is the semi-mechanistic model? Answer In the last few years the semi-mechanistic model which is based on key embrittlement mechanisms of chemical elements influence is proposed. The model includes three basic mechanisms influencing to radiation embrittlement for RPV steel ...
Birkhoff`s Theorem
... a contradiction. Proof of Birkhoff ’s theorem: We proceed by induction on the number of nonzero entries in the matrix. Let M0 be a doubly stochastic matrix. By the key lemma, the associated graph has a perfect matching. Underline the entries associated to the edges in the matching. For example in th ...
... a contradiction. Proof of Birkhoff ’s theorem: We proceed by induction on the number of nonzero entries in the matrix. Let M0 be a doubly stochastic matrix. By the key lemma, the associated graph has a perfect matching. Underline the entries associated to the edges in the matching. For example in th ...
Overview Quick review The advantages of a diagonal matrix
... The goal in this section is to develop a useful factorisation A = PDP −1 , for an n × n matrix A. This factorisation has several advantages: it makes transparent the geometric action of the associated linear transformation, and it permits easy calculation of Ak for large values of k: ...
... The goal in this section is to develop a useful factorisation A = PDP −1 , for an n × n matrix A. This factorisation has several advantages: it makes transparent the geometric action of the associated linear transformation, and it permits easy calculation of Ak for large values of k: ...
2.3 Characterizations of Invertible Matrices Theorem 8 (The
... 2.3 Characterizations of Invertible Matrices Theorem 8 (The Invertible Matrix Theorem) Let A be a square n × n matrix. The the following statements are equivalent (i.e., for a given A, they are either all true or all false). a. A is an invertible matrix. b. A is row equivalent to I n . c. A has n pi ...
... 2.3 Characterizations of Invertible Matrices Theorem 8 (The Invertible Matrix Theorem) Let A be a square n × n matrix. The the following statements are equivalent (i.e., for a given A, they are either all true or all false). a. A is an invertible matrix. b. A is row equivalent to I n . c. A has n pi ...
EET 465 LAB #2 - Pui Chor Wong
... code, is a linear combination of a set of k basis vectors, each of length n, denoted by g1, ...
... code, is a linear combination of a set of k basis vectors, each of length n, denoted by g1, ...
Revision 08/01/06
... for matrices. For example, in multiplication, all non-zero real numbers have a multiplicative inverse, where only a select set of matrices have a multiplicative inverse. What makes the process of teaching matrix multiplication different is the Sitn22MtrxOper060801.doc ...
... for matrices. For example, in multiplication, all non-zero real numbers have a multiplicative inverse, where only a select set of matrices have a multiplicative inverse. What makes the process of teaching matrix multiplication different is the Sitn22MtrxOper060801.doc ...
Procrustes distance
... Procrustes space. As written here, the routine will print pairwise distances in the SAS OUTPUT window; distances will also be stored in a dataset named “procd”. The matrix algebra for Orthogonal Procrustes Analysis comes from Rohlf & Slice (1990); the code here follows their notation as closely as p ...
... Procrustes space. As written here, the routine will print pairwise distances in the SAS OUTPUT window; distances will also be stored in a dataset named “procd”. The matrix algebra for Orthogonal Procrustes Analysis comes from Rohlf & Slice (1990); the code here follows their notation as closely as p ...
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.