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单词 ProofOfDeterminantLowerBoundOfAStrictDiagonallyDominantMatrix
释义

proof of determinant lower bound of a strict diagonally dominant matrix


Let’s define, for any i=1,2,,n

hi=|aii|-j=1,ji|aij|

Then, by strict diagonally dominance, one has hi>0  i.Let D=diag{(h1)-1,(h2)-1,,(hn)-1} and B=DA, so that the i-th row of B matrix is equal to the corresponding row of A matrix multiplied by (hi)-1. In this way , one has

di=|bii|-j=1,ji|bij|
=|aii|hi-j=1,ji|aij|hi
=1

Now, let λ be an eigenvalueMathworldPlanetmathPlanetmathPlanetmathPlanetmath of B, and v=[v1,v2,,vn]the corresponding eigenvectorMathworldPlanetmathPlanetmathPlanetmath; let moreover p be the index of the maximalcomponentPlanetmathPlanetmathPlanetmath of v, i.e.

|vp||vi| i

Of course, by definition of eigenvector, |vp|>0.Writing the p-th characteristic equationMathworldPlanetmathPlanetmath, we have:

λvp=j=1nbpjvj
=bppvp+j=1,jpnbpjvj

so that, being |vjvp|1,

λ=bpp+j=1,jpnbpjvjvp
|λ|=|bpp+j=1,jpnbpjvjvp|
||bpp|-|j=1,jpnbpjvjvp||
||bpp|-j=1,jpn|bpj||vjvp|| (*)
||bpp|-j=1,jpn|bpj|| (**)
=|bpp|-j=1,jpn|bpj|
=dp=1

In this way, we found that each eigenvalue of B is greater than one inabsolute valueMathworldPlanetmathPlanetmathPlanetmath; for this reason,

|det(B)|=|i=1nλi|1

Finally,

det(D)=i=1n(hi)-1=(i=1nhi)-1

so that

1|det(B)|
=|det(D)||det(A)|
=(i=1nhi)-1|det(A)|

whence the thesis.

Remark: Perhaps it could be not immediately evident where thehypothesis of strict diagonally dominance is employed in this proof; infact, inequality (*) and (**) would be, in a general case, not valid; they can be stated only because we can assure, by virtue of strictdiagonally dominance, that the final argument of the absolute value (|bpp|-j=1,jpn|bpj|) does remain positive.

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更新时间:2025/5/4 11:59:37