Properties of solutions to system of linear equations with diagnonally dominant, positive (definite) matrixConsider a system of linear equations A x = b, where A = [ 2 1 1 4 1 ⋱ ⋱ ⋱ 1 4 1 1 2 ] ∈ R n × n and n ≥ 3I would like to prove the following two statements, where the inequalities are element-wise: b ≥ 0 ⟹ x ≥ 0 b &gt; 0 ⟹ x &gt; 0 .

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Properties of solutions to system of linear equations with diagnonally dominant, positive (definite) matrixConsider a system of linear equations   A  x  =  b, where                    A                            =                              [                                                    2                                            1                                                                                                              1                                            4                                            1                                                                                                              ⋱                                            ⋱                                            ⋱                                                                                                              1                                            4                                            1                                                                                                              1                                            2                                              ]                ∈                          R                      n            ×            n                              and   n  ≥  3I would like to prove the following two statements, where the inequalities are element-wise:                    b        ≥        0                                                        ⟹                                                        x        ≥        0                            b        &amp;gt;        0                            ⟹                    x        &amp;gt;        0                 .

Haggar72 · Accepted Answer

You are essentially trying to prove that the inverse of   A is a positive matrix, which is not true as you can check by computing the inverse for, say,   n  =  3:You can use the inverse to directly find a counterexample: when  b  =      [                            1                                      0                                      0                      ]  we have  x  =      [                                        7            12                                                                          −              1                        6                                                            1            12                                ]    ≱  0.

Properties of solutions to system of linear equations with diagnonally dominant, positive (definite)

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