TY - JOUR AU - Dragomir, Silvestru Sever PY - 2022/12/09 Y2 - 2024/03/28 TI - Desigualdades determinantes para matrices definidas positivas a través de desigualdades young aditivas y multiplicativas JF - Revista Integración, temas de matemáticas JA - Rev. Integr. temas mat. VL - 40 IS - 2 SE - Artículo Original DO - 10.18273/revint.v40n2-2022004 UR - https://revistas.uis.edu.co/index.php/revistaintegracion/article/view/13980 SP - 193-206 AB - <p><span dir="ltr" role="presentation">En este trabajo demostramos entre otros que, si las matrices definidas positivas </span><span dir="ltr" role="presentation">A, B</span> <span dir="ltr" role="presentation">de orden</span> <span dir="ltr" role="presentation">n</span> <span dir="ltr" role="presentation">satisface la condición</span> <span dir="ltr" role="presentation">&lt; mI</span><span dir="ltr" role="presentation">n</span> <span dir="ltr" role="presentation">≤</span> <span dir="ltr" role="presentation">B</span> <span dir="ltr" role="presentation">−</span><span dir="ltr" role="presentation">A</span> <span dir="ltr" role="presentation">≤</span> <span dir="ltr" role="presentation">M I</span><span dir="ltr" role="presentation">n</span><span dir="ltr" role="presentation">, </span><span dir="ltr" role="presentation">para algunas constantes </span><span dir="ltr" role="presentation">0</span> <span dir="ltr" role="presentation">&lt; m &lt; M,</span> <span dir="ltr" role="presentation">donde</span> <span dir="ltr" role="presentation">I</span><span dir="ltr" role="presentation">n</span> <span dir="ltr" role="presentation">es la matriz identidad, entonces</span><br role="presentation" /><span dir="ltr" role="presentation">0</span> <span dir="ltr" role="presentation">≤</span> <span dir="ltr" role="presentation">(1</span> <span dir="ltr" role="presentation">−</span> <span dir="ltr" role="presentation">t</span><span dir="ltr" role="presentation">) [det (</span><span dir="ltr" role="presentation">A</span><span dir="ltr" role="presentation">)]</span><span dir="ltr" role="presentation">−</span><span dir="ltr" role="presentation">1</span> <span dir="ltr" role="presentation">+</span> <span dir="ltr" role="presentation">t</span> <span dir="ltr" role="presentation">[det (</span><span dir="ltr" role="presentation">A</span> <span dir="ltr" role="presentation">+</span> <span dir="ltr" role="presentation">mI</span><span dir="ltr" role="presentation">n</span><span dir="ltr" role="presentation">)]</span><span dir="ltr" role="presentation">−</span><span dir="ltr" role="presentation">1</span> <span dir="ltr" role="presentation">−</span> <span dir="ltr" role="presentation">[det (</span><span dir="ltr" role="presentation">A</span> <span dir="ltr" role="presentation">+</span> <span dir="ltr" role="presentation">mtI</span><span dir="ltr" role="presentation">n</span><span dir="ltr" role="presentation">)]</span><span dir="ltr" role="presentation">−</span><span dir="ltr" role="presentation">1</span><br role="presentation" /><span dir="ltr" role="presentation">≤</span> <span dir="ltr" role="presentation">(1</span> <span dir="ltr" role="presentation">−</span> <span dir="ltr" role="presentation">t</span><span dir="ltr" role="presentation">) [det (</span><span dir="ltr" role="presentation">A</span><span dir="ltr" role="presentation">)]</span><span dir="ltr" role="presentation">−</span><span dir="ltr" role="presentation">1</span> <span dir="ltr" role="presentation">+</span> <span dir="ltr" role="presentation">t</span> <span dir="ltr" role="presentation">[det (</span><span dir="ltr" role="presentation">B</span><span dir="ltr" role="presentation">)]</span><span dir="ltr" role="presentation">−</span><span dir="ltr" role="presentation">1</span> <span dir="ltr" role="presentation">−</span> <span dir="ltr" role="presentation">[det ((1</span> <span dir="ltr" role="presentation">−</span> <span dir="ltr" role="presentation">t</span><span dir="ltr" role="presentation">)</span> <span dir="ltr" role="presentation">A</span> <span dir="ltr" role="presentation">+</span> <span dir="ltr" role="presentation">tB</span><span dir="ltr" role="presentation">)]</span><span dir="ltr" role="presentation">−</span><span dir="ltr" role="presentation">1</span><br role="presentation" /><span dir="ltr" role="presentation">≤</span> <span dir="ltr" role="presentation">(1</span> <span dir="ltr" role="presentation">−</span> <span dir="ltr" role="presentation">t</span><span dir="ltr" role="presentation">) [det (</span><span dir="ltr" role="presentation">A</span><span dir="ltr" role="presentation">)]</span><span dir="ltr" role="presentation">−</span><span dir="ltr" role="presentation">1</span> <span dir="ltr" role="presentation">+</span> <span dir="ltr" role="presentation">t</span> <span dir="ltr" role="presentation">[det (</span><span dir="ltr" role="presentation">A</span> <span dir="ltr" role="presentation">+</span> <span dir="ltr" role="presentation">M I</span><span dir="ltr" role="presentation">n</span><span dir="ltr" role="presentation">)]</span><span dir="ltr" role="presentation">−</span><span dir="ltr" role="presentation">1</span> <span dir="ltr" role="presentation">−</span> <span dir="ltr" role="presentation">[det (</span><span dir="ltr" role="presentation">A</span> <span dir="ltr" role="presentation">+</span> <span dir="ltr" role="presentation">M tI</span><span dir="ltr" role="presentation">n</span><span dir="ltr" role="presentation">)]</span><span dir="ltr" role="presentation">−</span><span dir="ltr" role="presentation">1</span><span dir="ltr" role="presentation">,</span><br role="presentation" /><span dir="ltr" role="presentation">para todo</span> <span dir="ltr" role="presentation">t</span> <span dir="ltr" role="presentation">∈</span> <span dir="ltr" role="presentation">[0</span><span dir="ltr" role="presentation">,</span> <span dir="ltr" role="presentation">1].</span></p> ER -