<abstract xmlns="http://www.w3.org/1999/xhtml">

In this paper we prove among others that, if (<italic>Aj</italic>)<italic>j</italic>=1,...,<italic>m</italic> are positive definite matrices of order <italic>n</italic> ≥ 2 and <italic>qj</italic> ≥ 0, <italic>j</italic> = 1, ..., <italic>m</italic> with <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_awutm-2023-0003_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="M1"><mstyle displaystyle="true"><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><mrow><msub><mi>q</mi><mi>j</mi></msub><mo>=</mo><mn>1</mn></mrow></mstyle></math><tex-math>$$\sum\nolimits_{j = 1}^m {{q_j} = 1} $$</tex-math></alternatives></inline-formula>, then
<disp-formula id="j_awutm-2023-0003_eq_002"><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_awutm-2023-0003_eq_002.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" id="M2"><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow><mo>≤</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>−</mo><msub><mrow><mi>min</mi></mrow><mrow><mi>i</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>m</mi><mo>}</mo></mrow></msub><mrow><mo>{</mo> <mrow><msub><mi>q</mi><mi>i</mi></msub></mrow> <mo>}</mo></mrow></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mo>×</mo></mtd><mtd columnalign="left"><mrow><mrow><mo>[</mo> <mrow><munderover><mrow><msup><mo>∑</mo><mtext>​</mtext></msup></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><msub><mi>q</mi><mi>i</mi></msub><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msub><mi>q</mi><mi>i</mi></msub></mrow><mo>)</mo></mrow><msup><mrow><mrow><mo>[</mo> <mrow><mi>det</mi><mrow><mo>(</mo><mrow><msub><mi>A</mi><mi>i</mi></msub></mrow><mo>)</mo></mrow></mrow> <mo>]</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><msup><mn>2</mn><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><munder><mrow><msup><mo>∑</mo><mtext>​</mtext></msup></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>&lt;</mo><mi>j</mi><mo>≤</mo><mi>m</mi></mrow></munder><msub><mi>q</mi><mi>i</mi></msub><msub><mi>q</mi><mi>j</mi></msub><msup><mrow><mrow><mo>[</mo> <mrow><mi>det</mi><mrow><mo>(</mo><mrow><msub><mi>A</mi><mi>i</mi></msub><mo>+</mo><msub><mi>A</mi><mi>j</mi></msub></mrow><mo>)</mo></mrow></mrow> <mo>]</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow> <mo>]</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mo>≤</mo></mtd><mtd columnalign="left"><mrow><munderover><mrow><msup><mo>∑</mo><mtext>​</mtext></msup></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><msub><mi>q</mi><mi>i</mi></msub><msup><mrow><mrow><mo>[</mo> <mrow><mi>det</mi><mrow><mo>(</mo><mrow><msub><mi>A</mi><mi>i</mi></msub></mrow><mo>)</mo></mrow></mrow> <mo>]</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><msup><mrow><mrow><mo>[</mo> <mrow><mi>det</mi><mrow><mo>(</mo><mrow><munderover><mrow><msup><mo>∑</mo><mtext>​</mtext></msup></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><msub><mi>q</mi><mi>i</mi></msub><msub><mi>A</mi><mi>i</mi></msub></mrow><mo>)</mo></mrow></mrow> <mo>]</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mo>≤</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mrow><msub><mrow><mi>min</mi></mrow><mrow><mi>i</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>m</mi><mo>}</mo></mrow></msub><mrow><mo>{</mo> <mrow><msub><mi>q</mi><mi>i</mi></msub></mrow> <mo>}</mo></mrow></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mo>×</mo></mtd><mtd columnalign="left"><mrow><mrow><mo>[</mo> <mrow><munderover><mrow><msup><mo>∑</mo><mtext>​</mtext></msup></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><msub><mi>q</mi><mi>i</mi></msub><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msub><mi>q</mi><mi>i</mi></msub></mrow><mo>)</mo></mrow><msup><mrow><mrow><mo>[</mo> <mrow><mi>det</mi><mrow><mo>(</mo><mrow><msub><mi>A</mi><mi>i</mi></msub></mrow><mo>)</mo></mrow></mrow> <mo>]</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><msup><mn>2</mn><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><munder><mrow><msup><mo>∑</mo><mtext>​</mtext></msup></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>&lt;</mo><mi>j</mi><mo>≤</mo><mi>m</mi></mrow></munder><msub><mi>q</mi><mi>i</mi></msub><msub><mi>q</mi><mi>j</mi></msub><msup><mrow><mrow><mo>[</mo> <mrow><mi>det</mi><mrow><mo>(</mo><mrow><msub><mi>A</mi><mi>i</mi></msub><mo>+</mo><msub><mi>A</mi><mi>j</mi></msub></mrow><mo>)</mo></mrow></mrow> <mo>]</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow> <mo>]</mo></mrow><mo>.</mo></mrow></mtd></mtr></mtable></math>
<tex-math><?CDATA \begin{eqnarray}0 &#x0026; \le \frac{1}{1-{\min }_{i\in \{1,\ldots ,m\}}\{{q}_{i}\}}\\ \times &#x0026; [\underset{i=1}{\overset{m}{{\sum }^{\text{}}}}{q}_{i}(1-{q}_{i}){[\det ({A}_{i})]}^{-1}-{2}^{n+1}\mathop{{\sum }^{\text{}}}\limits_{1\le i\lt j\le m}{q}_{i}{q}_{j}{[\det ({A}_{i}+{A}_{j})]}^{-1}]\\ \le &#x0026; \underset{i=1}{\overset{m}{{\sum }^{\text{}}}}{q}_{i}{[\det ({A}_{i})]}^{-1}-{[\det (\underset{i=1}{\overset{m}{{\sum }^{\text{}}}}{q}_{i}{A}_{i})]}^{-1}\\ \le &#x0026; \frac{1}{{\min }_{i\in \{1,\ldots ,m\}}\{{q}_{i}\}}\\ \times &#x0026; [\underset{i=1}{\overset{m}{{\sum }^{\text{}}}}{q}_{i}(1-{q}_{i}){[\det ({A}_{i})]}^{-1}-{2}^{n+1}\mathop{{\sum }^{\text{}}}\limits_{1\le i\lt j\le m}{q}_{i}{q}_{j}{[\det ({A}_{i}+{A}_{j})]}^{-1}].\end{eqnarray}?></tex-math></alternatives></disp-formula>
</abstract>

In this paper we prove among others that, if (Aj)j=1,...,m are positive definite matrices of order n ≥ 2 and qj ≥ 0, j = 1, ..., m with ∑j=1mqj=1$$\sum\nolimits_{j = 1}^m {{q_j} = 1} $$, then
0≤11−mini∈{1,…,m}{ qi }×[ ∑​i=1mqi(1−qi)[ det(Ai) ]−1−2n+1∑​1≤i<j≤mqiqj[ det(Ai+Aj) ]−1 ]≤∑​i=1mqi[ det(Ai) ]−1−[ det(∑​i=1mqiAi) ]−1≤1mini∈{1,…,m}{ qi }×[ ∑​i=1mqi(1−qi)[ det(Ai) ]−1−2n+1∑​1≤i<j≤mqiqj[ det(Ai+Aj) ]−1 ].

Determinant Inequalities for Positive Definite Matrices Via Diananda’s Result for Arithmetic and Geometric Weighted Means

Annals of West University of Timisoara - Mathematics and Computer Science

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

In this paper we prove among others that, if (Aj)j=1,...,m are positive definite matrices of order n ≥ 2 and qj ≥ 0, j = 1, ..., m with...

Determinant Inequalities for Positive Definite Matrices Via Diananda’s Result for Arithmetic and Geometric Weighted Means

Publié en ligne: 03 mai 2023

Pages: 21 - 34

DOI: https://doi.org/10.2478/awutm-2023-0003

Mots clés
Positive definite matrices, Determinants, Inequalities

© 2023 Silvestru Sever Dragomir, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Determinant Inequalities for Positive Definite Matrices Via Diananda’s Result for Arithmetic and Geometric Weighted Means

Publié en ligne: 03 mai 2023

Pages: 21 - 34

DOI: https://doi.org/10.2478/awutm-2023-0003

Mots clésPositive definite matrices, Determinants, Inequalities

© 2023 Silvestru Sever Dragomir, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Mots clés
Positive definite matrices, Determinants, Inequalities