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This makes calculating the matrix elements of the interaction Hamiltonian very important for finding the energy levels and wave-functions of particles in a different atomic elements and nuclide. This many-body Hamiltonian problem becomes very complicated with the addition of more electrons, protons and neutrons as we go to other elements in the ...
Signature matrix: A diagonal matrix where the diagonal elements are either +1 or −1. Single-entry matrix: A matrix where a single element is one and the rest of the elements are zero. Skew-Hermitian matrix: A square matrix which is equal to the negative of its conjugate transpose, A * = −A. Skew-symmetric matrix
A matrix is diagonal if and only if it is both upper-and lower-triangular. A diagonal matrix is symmetric. The identity matrix I n and zero matrix are diagonal. A 1×1 matrix is always diagonal. The square of a 2×2 matrix with zero trace is always diagonal.
In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix is the list of entries , where =. All off-diagonal elements are zero in a diagonal matrix. The following four matrices have their main diagonals indicated by red ones:
A hollow matrix may be a square matrix whose diagonal elements are all equal to zero. [3] That is, an n × n matrix A = (a ij) is hollow if a ij = 0 whenever i = j (i.e. a ii = 0 for all i). The most obvious example is the real skew-symmetric matrix. Other examples are the adjacency matrix of a finite simple graph, and a distance matrix or ...
A matrix that is both upper and lower triangular is diagonal. Matrices that are similar to triangular matrices are called triangularisable. A non-square (or sometimes any) matrix with zeros above (below) the diagonal is called a lower (upper) trapezoidal matrix. The non-zero entries form the shape of a trapezoid.
Furthermore, if a real tridiagonal matrix A satisfies a k,k+1 a k+1,k > 0 for all k, so that the signs of its entries are symmetric, then it is similar to a Hermitian matrix, by a diagonal change of basis matrix. Hence, its eigenvalues are real. If we replace the strict inequality by a k,k+1 a k+1,k ≥ 0, then by continuity, the eigenvalues ...
Since A(t) has a classical Fourier series with only the lowest frequency, and the matrix element A mn is the (m − n) th Fourier coefficient of the classical orbit, the matrix for A is nonzero only on the line just above the diagonal, where it is equal to √ 2E n.