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Volume 12, Issue 1
The Solvability Conditions for the Inverse Problem of Matrices Positive Semidefinite on a Subspace

Xi-Yan Hu, Lei Zhang & Wei-Zhang Du

J. Comp. Math., 12 (1994), pp. 78-87.

Published online: 1994-12

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  • Abstract

This paper studies the following two problem:
Problem Ⅰ. Given $X,B∈R^{n×m}$, find $A∈P_{s,n}$, such that $AX=B$, where
$P_{s,n}$={$A∈SR^{n×n}|x^T AX≥0，∀S^Tx=0$ , for given $S∈R^{n×p}_p$}.
Problem Ⅱ. Given $A^*∈R^{n×n}$, find $\hat{A}∈S_E$, such that $||A^*-\hat{A}||$=inf$_{A∈S_E}||A^*-A||$ where $S_E$ denotes the solutions set of  Problem Ⅰ.
The necessary and sufficient conditions for the solvability of Problem Ⅰ, the expression of the general solution of Problem Ⅰ and the solution of Problem Ⅱ are given for two case. For the general case, the equivalent form of conditions for the solvability of Problem  Ⅰ is given.

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@Article{JCM-12-78, author = {Hu , Xi-YanZhang , Lei and Du , Wei-Zhang}, title = {The Solvability Conditions for the Inverse Problem of Matrices Positive Semidefinite on a Subspace}, journal = {Journal of Computational Mathematics}, year = {1994}, volume = {12}, number = {1}, pages = {78--87}, abstract = {

This paper studies the following two problem:
Problem Ⅰ. Given $X,B∈R^{n×m}$, find $A∈P_{s,n}$, such that $AX=B$, where
$P_{s,n}$={$A∈SR^{n×n}|x^T AX≥0，∀S^Tx=0$ , for given $S∈R^{n×p}_p$}.
Problem Ⅱ. Given $A^*∈R^{n×n}$, find $\hat{A}∈S_E$, such that $||A^*-\hat{A}||$=inf$_{A∈S_E}||A^*-A||$ where $S_E$ denotes the solutions set of  Problem Ⅰ.
The necessary and sufficient conditions for the solvability of Problem Ⅰ, the expression of the general solution of Problem Ⅰ and the solution of Problem Ⅱ are given for two case. For the general case, the equivalent form of conditions for the solvability of Problem  Ⅰ is given.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/10229.html} }
TY - JOUR T1 - The Solvability Conditions for the Inverse Problem of Matrices Positive Semidefinite on a Subspace AU - Hu , Xi-Yan AU - Zhang , Lei AU - Du , Wei-Zhang JO - Journal of Computational Mathematics VL - 1 SP - 78 EP - 87 PY - 1994 DA - 1994/12 SN - 12 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/10229.html KW - AB -

This paper studies the following two problem:
Problem Ⅰ. Given $X,B∈R^{n×m}$, find $A∈P_{s,n}$, such that $AX=B$, where
$P_{s,n}$={$A∈SR^{n×n}|x^T AX≥0，∀S^Tx=0$ , for given $S∈R^{n×p}_p$}.
Problem Ⅱ. Given $A^*∈R^{n×n}$, find $\hat{A}∈S_E$, such that $||A^*-\hat{A}||$=inf$_{A∈S_E}||A^*-A||$ where $S_E$ denotes the solutions set of  Problem Ⅰ.
The necessary and sufficient conditions for the solvability of Problem Ⅰ, the expression of the general solution of Problem Ⅰ and the solution of Problem Ⅱ are given for two case. For the general case, the equivalent form of conditions for the solvability of Problem  Ⅰ is given.

Hu , Xi-YanZhang , Lei and Du , Wei-Zhang. (1994). The Solvability Conditions for the Inverse Problem of Matrices Positive Semidefinite on a Subspace. Journal of Computational Mathematics. 12 (1). 78-87. doi:
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