1.1 Singular values and matrix inversion For non-symmetric matrices, the eigenvalues and singular values are not equivalent. However, they share one important property: Fact 1 A matrix A, NxN, is invertible iff all of its singular values are non-zero.

We consider computing the singular value decomposition of a bidiagonal matrix B. This problem arises in the singular value decomposition of a general matrix, and in the eigenproblem for a symmetric positive de nite tridiagonal matrix. We show that if the entries of B are known with high relative accuracy, the singular values and singular vectors of B will be determined to much higher accuracy t...

Neutrosophy is a new branch of philosophy that studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra. Aim of this article is to find the maximum and minimum solution of the fuzzy neutrosophic soft relational equations xA=b and Ax=b, where x and b are fuzzy neutrosophic soft vector and A is a fuzzy neutrosophic soft matrix. Wheneve...

The classical perturbation theory for Hermitian matrix eigenvalue and singular value problems provides bounds on the absolute differences between approximate eigenvalues (singular values) and the true eigenvalues (singular values) of a matrix. These bounds may be bad news for small eigenvalues (singular values), which thereby suffer worse relative uncertainty than large ones. However, there are...

H altitude or impulse response matrix I identity matrix j dummy index M Mach number R correlation coefficient S matrix of singular values STBL left stabiliser position STBR right stabiliser position t, Δt time, time step T projected state matrix multiplied by the instrument TEFL left trailing-edge flap deflection TEFR right trailing-edge flap deflection u system input vector U matrix of past or...

In this paper, we study the matrix denoising model $Y=S+X$, where $S$ is a low rank deterministic signal and $X$ random noise matrix, both are $M\times n$. scenario that $M$ $n$ comparably large signals supercritical, fluctuation of outlier singular vectors $Y$, under fully general assumptions on structure distribution $X$. More specifically, derive limiting angles between principal $Y$ their c...

A class of matrix valued functions defined by singular values of nonsymmetric matrices are shown to have many properties analogous to matrix valued functions defined by eigenvalues of symmetric matrices. In particular, the (smoothed) matrix valued Fischer-Burmeister function is proved to be strongly semismooth everywhere. This result is also used to show the strong semismoothness of the (smooth...

We propose new algorithms for singular value decomposition (SVD) of very large-scale matrices based on a low-rank tensor approximation technique called the tensor train (TT) format. The proposed algorithms can compute several dominant singular values and corresponding singular vectors for large-scale structured matrices given in a TT format. The computational complexity of the proposed methods ...