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A complex symmetric matrix can be 'diagonalized' using a unitary matrix: thus if is a complex symmetric matrix, there is a unitary matrix such that is a real diagonal matrix with non-negative entries.
Matrix inversion is the process of finding the matrix which when multiplied by the original matrix gives the identity matrix. [ 2 ] Over a field , a square matrix that is not invertible is called singular or degenerate .
The Laplacian matrix of a directed graph is by definition generally non-symmetric, while, e.g., traditional spectral clustering is primarily developed for undirected graphs with symmetric adjacency and Laplacian matrices. A trivial approach to apply techniques requiring the symmetry is to turn the original directed graph into an undirected ...
If instead, A is equal to the negative of its transpose, that is, A = −A T, then A is a skew-symmetric matrix. In complex matrices, symmetry is often replaced by the concept of Hermitian matrices, which satisfies A ∗ = A, where the star or asterisk denotes the conjugate transpose of the matrix, that is, the transpose of the complex ...
A matrix with relatively few non-zero elements. Sparse matrix algorithms can tackle huge sparse matrices that are utterly impractical for dense matrix algorithms. Symmetric matrix: A square matrix which is equal to its transpose, A = A T (a i,j = a j,i). Toeplitz matrix: A matrix with constant diagonals. Totally positive matrix
Examples of passive components deliberately designed to be non-reciprocal include circulators and isolators. [3] The transfer function of a reciprocal network has the property that it is symmetrical about the main diagonal if expressed in terms of a z-parameter, y-parameter, or s-parameter matrix. A non-symmetrical matrix implies a non ...
An idempotent matrix is always diagonalizable. [3] Its eigenvalues are either 0 or 1: if is a non-zero eigenvector of some idempotent matrix and its associated eigenvalue, then = = = = =, which implies {,}.
Let A be a square n × n matrix with n linearly independent eigenvectors q i (where i = 1, ..., n).Then A can be factored 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.