20.500.12556/RUL-173614 Matrične različice neenakosti med aritmetično in geometrijsko sredino delo diplomskega seminarja Matrix versions of the inequality between arithmetic and geometric means Aritmetično sredino pozitivno definitnih matrik $A$ in $B$ smo naravno definirali kot $M_A = \frac{1}{2}(A + B)$, njuno geometrijsko sredino pa kot $M_G = A^\frac{1}{2}(A^{-\frac{1}{2}} B A^{-\frac{1}{2}} )^\frac{1}{2} A ^\frac{1}{2}.$ V diplomskem delu bomo dokazali bistvene lastnosti geometrijske sredine in pokazali, da med sredinama velja zveza $M_G \leq M_A,$ kar pomeni, da je matrika $M_A - M_G$ pozitivno semidefinitna. Pri kompleksnih matrikah bomo spoznali in dokazali dve različici neenakosti. Prva velja za $A, B \in \mathbb{C}^{n \times n}$ in se glasi $$s_j (A^* B) \le \frac{1}{2} s_j (AA^* + BB^*), $$ pri čemer je $s_j$ $j$-ta največja singularna vrednost matrike, $j = 1, \dots, n.$ Druga različica velja za poljubne matrike $A, B, X$ in pravi $$ s_1(A^*XB) \leq \tfrac{1}{2} s_1(AA^*X + XBB^*). $$ We naturally define the arithmetic mean of two positive definite matrices $A$ and $B$ as $M_A = \frac{1}{2}(A + B)$ and their geometric mean as $M_G = A^{\frac{1}{2}} \left(A^{-\frac{1}{2}} B A^{-\frac{1}{2}} \right)^{\frac{1}{2}} A^{\frac{1}{2}}.$ In the thesis, we will prove essential properties of the geometric mean and we will show that $M_G \leq M_A,$ which means that the matrix $M_A-M_G$ is positive semidefinite. For complex matrices, we will explore and prove two versions of the inequality. The first holds for $A, B \in \mathbb{C}^{n \times n}$ and states $$s_j(A^* B) \le \frac{1}{2} s_j(AA^* + BB^*),$$ where $s_j$ denotes $j$-th largest singular value of the matrix, $j = 1, \dots, n.$ The second version holds for arbitrary matrices $ A, B, X$ and states $$ s_1(A^*XB) \leq \tfrac{1}{2} s_1(AA^*X + XBB^*). $$ aritmetična sredina matrik geometrijska sredina matrik Löwnerjeva delna urejenost matrične neenakosti pozitivna (semi)definitnost spektralni radij 2-norma arithmetic mean of matrices geometric mean of matrices Löwner partial order matrix inequalities positive (semi)definiteness spectral radius 2-norm true false false Slovenski jezik Angleški jezik Delo diplomskega seminarja/zaključno seminarsko delo/naloga 2025-09-19 08:15:12 2025-09-19 08:15:16 2025-11-11 03:47:45 0000-00-00 00:00:00 2025 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 512 153980 250124803 19048.pdf 19048.pdf 1 29AE5B9326CF6BFF48FDC7114135858D c3a3912fa853c0148ce69d06012da0b1c880d7b3c167f0dcb3278eb2e0a94645 d59e5e22-951f-11f0-9328-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=218220 Fakulteta za matematiko in fiziko 0 0 0