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In the matrix notation, the adjacency matrix of the undirected graph could, e.g., be defined as a Boolean sum of the adjacency matrix of the original directed graph and its matrix transpose, where the zero and one entries of are treated as logical, rather than numerical, values, as in the following example:
The fundamental fact about diagonalizable maps and matrices is expressed by the following: An matrix over a field is diagonalizable if and only if the sum of the dimensions of its eigenspaces is equal to , which is the case if and only if there exists a basis of consisting of eigenvectors of .
The name converse arises because the reversal of arrows corresponds to taking the converse of an implication in logic. The name transpose is because the adjacency matrix of the transpose directed graph is the transpose of the adjacency matrix of the original directed graph.
The transpose of an m-by-n matrix A is the n-by-m matrix A T (also denoted A tr or t A) formed by turning rows into columns and vice versa: (A T) i,j = A j,i. For ...
Let A be a square n × n matrix with n linearly independent eigenvectors q i (where i = 1, ..., n).Then A can be factorized as = where Q is the square n × n matrix whose i th column is the eigenvector q i of A, and Λ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, Λ ii = λ i.
Throughout this article, boldfaced unsubscripted and are used to refer to random vectors, and Roman subscripted and are used to refer to scalar random variables.. If the entries in the column vector = (,, …,) are random variables, each with finite variance and expected value, then the covariance matrix is the matrix whose (,) entry is the covariance [1]: 177 ...
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j: = ¯
In mathematics, a Young tableau (/ t æ ˈ b l oʊ, ˈ t æ b l oʊ /; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus.It provides a convenient way to describe the group representations of the symmetric and general linear groups and to study their properties.