Lecture 2: Spectra of Graphs 1 Definitions
... Lemma 2.6. Consider any undirected graph G with adjacency matrix A. 1. If G is d-regular, then λ1 = d and |λi | ≤ d for i = 2, . . . , n. 2. G is connected iff λ2 < d, i.e., the eigenvalue d has multiplicity 1. Moreover, the number of connected components of G equals the multiplicity of eigenvalue d ...
... Lemma 2.6. Consider any undirected graph G with adjacency matrix A. 1. If G is d-regular, then λ1 = d and |λi | ≤ d for i = 2, . . . , n. 2. G is connected iff λ2 < d, i.e., the eigenvalue d has multiplicity 1. Moreover, the number of connected components of G equals the multiplicity of eigenvalue d ...
On the energy and spectral properties of the he matrix of hexagonal
... The most common algebraic representation of a graph is the adjacency matrix, followed by the Laplacian matrix, incidence matrix and various other forms [3], [4]. While capturing most structural information of graphs, these matrices ignore the orientation of edges in the graph. Generally, graphs do n ...
... The most common algebraic representation of a graph is the adjacency matrix, followed by the Laplacian matrix, incidence matrix and various other forms [3], [4]. While capturing most structural information of graphs, these matrices ignore the orientation of edges in the graph. Generally, graphs do n ...
M.4. Finitely generated Modules over a PID, part I
... zero ideal, hence free of rank 0. This verifies the case n = 1. Suppose that F has rank n > 1 and that for any free R module F ′ of rank less than n and for any submodule N ′ of F ′ , N ′ is free, and rank(N ′ ) ≤ rank(F ′ ). Let {f1 , . . . , fn } be a basis of F , put F ′ = span({f1 , . . . , fn−1 ...
... zero ideal, hence free of rank 0. This verifies the case n = 1. Suppose that F has rank n > 1 and that for any free R module F ′ of rank less than n and for any submodule N ′ of F ′ , N ′ is free, and rank(N ′ ) ≤ rank(F ′ ). Let {f1 , . . . , fn } be a basis of F , put F ′ = span({f1 , . . . , fn−1 ...
GigaTensor: Scaling Tensor Analysis Up By 100 Times
... tensors (having attracted best paper awards, e.g. see [20]). However, the toolboxes have critical restrictions: 1) they operate strictly on data that can fit in the main memory, and 2) their scalability is limited by the scalability of Matlab. In [4, 20], efficient ways of computing tensor decomposi ...
... tensors (having attracted best paper awards, e.g. see [20]). However, the toolboxes have critical restrictions: 1) they operate strictly on data that can fit in the main memory, and 2) their scalability is limited by the scalability of Matlab. In [4, 20], efficient ways of computing tensor decomposi ...
Removal Lemmas for Matrices
... Here a set of pairwise-disjoint A-copies in M is a set of s × t submatrices of M , all equal to A, such that any entry of M is contained in at most one of the submatrices. Theorem 1.2 is an analogue for binary matrices of the non-induced graph removal lemma. However, in the graph removal lemma, δ −1 ...
... Here a set of pairwise-disjoint A-copies in M is a set of s × t submatrices of M , all equal to A, such that any entry of M is contained in at most one of the submatrices. Theorem 1.2 is an analogue for binary matrices of the non-induced graph removal lemma. However, in the graph removal lemma, δ −1 ...
Determinants - ShawTLR.Net
... How to Find the Adjoint of a Matrix? The adjoint of a matrix can be found by taking the transpose of the matrix of cofactors from A. In our previous example, we have found the cofactors A11, A21, A31. If we continue to solve for the rest of the cofactors for matrix A, namely A12, A22, A32 , A13, A2 ...
... How to Find the Adjoint of a Matrix? The adjoint of a matrix can be found by taking the transpose of the matrix of cofactors from A. In our previous example, we have found the cofactors A11, A21, A31. If we continue to solve for the rest of the cofactors for matrix A, namely A12, A22, A32 , A13, A2 ...
Faculty of Engineering - Multimedia University
... similarly to eye and make matrices with elements equal to zero, elements equal to one, and random elements respectively. These commands can also be used to create nonsquare matrices. For example, zeros(2,4) generates a 2 x 4 matrix of zeros. Finally, A = A' command will give the tranpose of the matr ...
... similarly to eye and make matrices with elements equal to zero, elements equal to one, and random elements respectively. These commands can also be used to create nonsquare matrices. For example, zeros(2,4) generates a 2 x 4 matrix of zeros. Finally, A = A' command will give the tranpose of the matr ...
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.