Chap1
... Def. An (n × n) matrix A is said to be nonsingular or invertible if there exists a matrix B such that AB=BA=I. The matrix B is said to be a multiplicative inverse of A. And B is denoted by A-1. Warning: In general, AB≠BA. Matrix multiplication is not commutative. ...
... Def. An (n × n) matrix A is said to be nonsingular or invertible if there exists a matrix B such that AB=BA=I. The matrix B is said to be a multiplicative inverse of A. And B is denoted by A-1. Warning: In general, AB≠BA. Matrix multiplication is not commutative. ...
document
... have a Toeplitz structure, and efficient algorithms (Schur recursions) exist to factor such matrices or their inverse. Schur recursions can be generalized to apply to general Toeplitz matrices [1]. The computation of the inverse of a Toeplitz matrix goes via Gohberg/Semencul recursions [2]. The resu ...
... have a Toeplitz structure, and efficient algorithms (Schur recursions) exist to factor such matrices or their inverse. Schur recursions can be generalized to apply to general Toeplitz matrices [1]. The computation of the inverse of a Toeplitz matrix goes via Gohberg/Semencul recursions [2]. The resu ...
9. Change of basis/coordinates Theorem Let β and β be two ordered
... β 0 be ordered bases for V and let P be the change of coordinates from β 0 to β. Then [T ]β = P [T ]β 0 P −1. • Let T be a linear operator on a finite dimensional vector space V . Then for any ordered bases β and γ of V , [T ]β is similar to [T ]γ . ...
... β 0 be ordered bases for V and let P be the change of coordinates from β 0 to β. Then [T ]β = P [T ]β 0 P −1. • Let T be a linear operator on a finite dimensional vector space V . Then for any ordered bases β and γ of V , [T ]β is similar to [T ]γ . ...
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 0. Singular matrices are rare in the sense that if you pick a random square matrix, it will almost surely not be singular. While the most common case is that of matr ...
... A square matrix that is not invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is 0. Singular matrices are rare in the sense that if you pick a random square matrix, it will almost surely not be singular. While the most common case is that of matr ...
form Given matrix The determinant is indicated by
... together… this is your “DOWN” total. 4) Draw “Up” diagonals under each of the three 3-term “Up” diagonals. ...
... together… this is your “DOWN” total. 4) Draw “Up” diagonals under each of the three 3-term “Up” diagonals. ...
Lecture 3
... Assuming first that no row permutation is necessary, the upper triangular matrix A(n) = U is therefore obtained as A(n) = Tn−1A(n−1) = Tn−1Tn−2A(n−2) = · · · = = Tn−1Tn−2 · · · T1A(1) = ΛA where the matrix Λ = Tn−1Tn−2 · · · T1 is lower triangular as a product of l.t. matrices. It follows that ΛA = ...
... Assuming first that no row permutation is necessary, the upper triangular matrix A(n) = U is therefore obtained as A(n) = Tn−1A(n−1) = Tn−1Tn−2A(n−2) = · · · = = Tn−1Tn−2 · · · T1A(1) = ΛA where the matrix Λ = Tn−1Tn−2 · · · T1 is lower triangular as a product of l.t. matrices. It follows that ΛA = ...
Projection on the intersection of convex sets
... point z ∗ , and that therefore we can use the semi-smooth Newton algorithm for computation of the projection point. The computation is now done in parallel and the number of iterations is drastically reduced comparing to existing algorithms. However, there is a time-consuming operation involved in t ...
... point z ∗ , and that therefore we can use the semi-smooth Newton algorithm for computation of the projection point. The computation is now done in parallel and the number of iterations is drastically reduced comparing to existing algorithms. However, there is a time-consuming operation involved in t ...
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.