The Bounds for Spectral Norm and Frobenius Norm Condition Number of a Simple Matrix
2
Citation
4
Reference
10
Related Paper
Citation Trend
Abstract:
In this paper, we give the estimations both of spectral and Frobenius norm condition number of a simple matrix. The estimations can be used to measure the sensitivity of the solution of linear systems.Keywords:
Matrix norm
Condition number
Matrix (chemical analysis)
Matrix algebra
Spectral measure
In this paper, we give the estimations both of spectral and Frobenius norm condition number of a simple matrix. The estimations can be used to measure the sensitivity of the solution of linear systems.
Matrix norm
Condition number
Matrix (chemical analysis)
Matrix algebra
Spectral measure
Cite
Citations (2)
The condition number of solutions to full rank linear least-squares problem are shown to be given by an optimization problem that involves nuclear norms of rank 2 matrices. The condition number is with respect to the least-squares coefficient matrix and 2-norms. It depends on three quantities each of which can contribute ill-conditioning. The literature presents several estimates for this condition number with varying results; even standard reference texts contain serious overestimates. The use of the nuclear norm affords a single derivation of the best known lower and upper bounds on the condition number and shows why there is unlikely to be a closed formula.
Rank (graph theory)
Condition number
Matrix norm
Least-squares function approximation
Matrix (chemical analysis)
Coefficient matrix
Cite
Citations (0)
Matrix norm
Low-rank approximation
Singular value
Rank (graph theory)
Matrix (chemical analysis)
Condition number
Matrix Completion
Cite
Citations (4)
The condition number of solutions to full rank linear least-squares problem are shown to be given by an optimization problem that involves nuclear norms of rank 2 matrices. The condition number is with respect to the least-squares coefficient matrix and 2-norms. It depends on three quantities each of which can contribute ill-conditioning. The literature presents several estimates for this condition number with varying results; even standard reference texts contain serious overestimates. The use of the nuclear norm affords a single derivation of the best known lower and upper bounds on the condition number and shows why there is unlikely to be a closed formula.
Rank (graph theory)
Matrix norm
Condition number
Least-squares function approximation
Matrix (chemical analysis)
Cite
Citations (0)
In this article, new upper and lower bounds for the spectral condition number are obtained. These bounds are constructed based on the Frobenius norm of some matrices related to the given matrix and its inverse. Hence, unlike some existing bounds, these new bounds are smooth functions with respect to the elements in the matrix. It is theoretically established that the new bounds are also sandwiched by the true value of the spectral condition number and its estimates using the Frobenius norms. Moreover, the bounds give the exact value of the spectral condition number when the matrix is unitary or of order less than 3. The new upper bound provided, via statistical numerical comparison, is shown to be the best when compared with existing results.
Matrix norm
Condition number
Matrix (chemical analysis)
Value (mathematics)
Singular value
Cite
Citations (10)
Matrix norm
Condition number
Matrix (chemical analysis)
Linear equation
Cite
Citations (12)
Condition number
Matrix norm
Matrix (chemical analysis)
Complex matrix
Cite
Citations (1)
We consider here the linear least squares problem $\min_{y \in \mathbb{R}^n}\|Ay-b\|_2$, where $b \in \mathbb{R}^m$ and $A \in \mathbb{R}^{m\times n}$ is a matrix of full column rank n, and we denote x its solution. We assume that both A and b can be perturbed and that these perturbations are measured using the Frobenius or the spectral norm for A and the Euclidean norm for b. In this paper, we are concerned with the condition number of a linear function of x ($L^Tx$, where $L \in \mathbb{R}^{n\times k}$) for which we provide a sharp estimate that lies within a factor $\sqrt{3}$ of the true condition number. Provided the triangular R factor of A from $A^TA=R^TR$ is available, this estimate can be computed in $2kn^2$ flops. We also propose a statistical method that estimates the partial condition number by using the exact condition numbers in random orthogonal directions. If R is available, this statistical approach enables us to obtain a condition estimate at a lower computational cost. In the case of the Frobenius norm, we derive a closed formula for the partial condition number that is based on the singular values and the right singular vectors of the matrix A.
Condition number
Matrix norm
Singular value
Rank (graph theory)
Matrix (chemical analysis)
Least-squares function approximation
Cite
Citations (43)
Matrix norm
Condition number
Singular value
Matrix (chemical analysis)
Rank (graph theory)
Cite
Citations (0)