In numerical linear algebra, the problem of computing the distance of a given matrix A from a given set P arises in different fields of matrix and control theory, where it is used to characterize the robustness of considered systems. Some examples include, but are not limited to, distance to singularity, matrix stability, measures in control theory, etc. (see e.g. [28, 33, 44, 45, 46, 53]). The problem consists in computing an element B ∈ P such that the distance between B and A is the smallest possible; under suitable assumptions on P, the problem is always well defined. The most common version in this matrix nearness context concerns the unstructured distance, which means that the optimization problem introduced for the computation of the matrix B does not take into account any specific structure of the original matrix A, e.g. its sparsity pattern, its reality, a particular design of the entries etc.. Quite recently, an increasing interest has risen for the structured version of the distance (see e.g. [21, 31, 34, 37, 38, 49, 65]), where the optimizer sought is required to preserve a specific structure that the matrix A has. In this case we talk about structured distance between A and the set P, which is clearly larger or equal than the unstructured distance between the same objects and it could have a different order of magnitude. Also in this case the problem may not be well-defined, for instance if the constraint forced by the structure is too strong and it makes impossible to find a matrix B ∈ P with the structure required, but again under reasonable assumptions it is possible to consider that the problem is solvable. The main motivation behind the introduction of the structured distance is that it allows to take into account some features of the original matrix A and it can be exploited to get a more appropriate matrix B that provides a more meaningful solution to the matrix nearness problem. In numerical linear algebra, the problem of computing the distance of a given matrix A from a given set P arises in different fields of matrix and control theory, where it is used to characterize the robustness of considered systems. Some examples include, but are not limited to, distance to singularity, matrix stability, measures in control theory, etc. (see e.g. [28, 33, 44, 45, 46, 53]). The problem consists in computing an element B ∈ P such that the distance between B and A is the smallest possible; under suitable assumptions on P, the problem is always well defined. The most common version in this matrix nearness context concerns the unstructured distance, which means that the optimization problem introduced for the computation of the matrix B does not take into account any specific structure of the original matrix A, e.g. its sparsity pattern, its reality, a particular design of the entries etc.. Quite recently, an increasing interest has risen for the structured version of the distance (see e.g. [21, 31, 34, 37, 38, 49, 65]), where the optimizer sought is required to preserve a specific structure that the matrix A has. In this case we talk about structured distance between A and the set P, which is clearly larger or equal than the unstructured distance between the same objects and it could have a different order of magnitude. Also in this case the problem may not be well-defined, for instance if the constraint forced by the structure is too strong and it makes impossible to find a matrix B ∈ P with the structure required, but again under reasonable assumptions it is possible to consider that the problem is solvable. The main motivation behind the introduction of the structured distance is that it allows to take into account some features of the original matrix A and it can be exploited to get a more appropriate matrix B that provides a more meaningful solution to the matrix nearness problem.

Low-rank properties in structured matrix nearness problems / Sicilia, Stefano. - (2025 Jan 13).

Low-rank properties in structured matrix nearness problems

SICILIA, STEFANO
2025-01-13

Abstract

In numerical linear algebra, the problem of computing the distance of a given matrix A from a given set P arises in different fields of matrix and control theory, where it is used to characterize the robustness of considered systems. Some examples include, but are not limited to, distance to singularity, matrix stability, measures in control theory, etc. (see e.g. [28, 33, 44, 45, 46, 53]). The problem consists in computing an element B ∈ P such that the distance between B and A is the smallest possible; under suitable assumptions on P, the problem is always well defined. The most common version in this matrix nearness context concerns the unstructured distance, which means that the optimization problem introduced for the computation of the matrix B does not take into account any specific structure of the original matrix A, e.g. its sparsity pattern, its reality, a particular design of the entries etc.. Quite recently, an increasing interest has risen for the structured version of the distance (see e.g. [21, 31, 34, 37, 38, 49, 65]), where the optimizer sought is required to preserve a specific structure that the matrix A has. In this case we talk about structured distance between A and the set P, which is clearly larger or equal than the unstructured distance between the same objects and it could have a different order of magnitude. Also in this case the problem may not be well-defined, for instance if the constraint forced by the structure is too strong and it makes impossible to find a matrix B ∈ P with the structure required, but again under reasonable assumptions it is possible to consider that the problem is solvable. The main motivation behind the introduction of the structured distance is that it allows to take into account some features of the original matrix A and it can be exploited to get a more appropriate matrix B that provides a more meaningful solution to the matrix nearness problem. In numerical linear algebra, the problem of computing the distance of a given matrix A from a given set P arises in different fields of matrix and control theory, where it is used to characterize the robustness of considered systems. Some examples include, but are not limited to, distance to singularity, matrix stability, measures in control theory, etc. (see e.g. [28, 33, 44, 45, 46, 53]). The problem consists in computing an element B ∈ P such that the distance between B and A is the smallest possible; under suitable assumptions on P, the problem is always well defined. The most common version in this matrix nearness context concerns the unstructured distance, which means that the optimization problem introduced for the computation of the matrix B does not take into account any specific structure of the original matrix A, e.g. its sparsity pattern, its reality, a particular design of the entries etc.. Quite recently, an increasing interest has risen for the structured version of the distance (see e.g. [21, 31, 34, 37, 38, 49, 65]), where the optimizer sought is required to preserve a specific structure that the matrix A has. In this case we talk about structured distance between A and the set P, which is clearly larger or equal than the unstructured distance between the same objects and it could have a different order of magnitude. Also in this case the problem may not be well-defined, for instance if the constraint forced by the structure is too strong and it makes impossible to find a matrix B ∈ P with the structure required, but again under reasonable assumptions it is possible to consider that the problem is solvable. The main motivation behind the introduction of the structured distance is that it allows to take into account some features of the original matrix A and it can be exploited to get a more appropriate matrix B that provides a more meaningful solution to the matrix nearness problem.
13-gen-2025
Low-rank properties in structured matrix nearness problems / Sicilia, Stefano. - (2025 Jan 13).
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.12571/33684
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