Applications of linear algebra in proving inequalities

U ovom radu je pokazano kako se poznavanjem osnovnih pojmova i činjenica iz linearne algebre mogu dokazati zanimljive, često i važne nejednakosti iz različitih matematičkih disciplina. Primjerima su ilustrirane neke primjene u algebri, kombinatorici, kombinatornoj geometriji te teoriji dizajna. Pri dokazivanju samih nejednakosti korištene su poznate činjenice poput svojstava ranga matrice, dimenzije potprostora vektorskih prostora, kriterij postojanja netrivijalnog rješenja homogenog sustava linearnih jednadžbi te Cauchy-Schwarzove nejednakosti. Zastupljen je i poopćeni oblik Binet-Cauchyjevog teorema o determinanti umnoška matrica, koji je takoder koristan u primjenama. Medu rezultatima dokazanima u radu obuhvaćene su Fisherova nejednakost za kombinatorne dizajne te gornja meda veličine ekviangularnog skupa pravaca u d-dimenzionalnom euklidskom prostoru, što pripada problemima kombinatorne geometrije koji već desetljećima zaokupljaju matematičare i o kojima su posljednjih godina postignuti neki značajni rezultati.In this paper, it is shown how some interesting, often important inequalities from various mathematical disciplines can be proved by using basic concepts and facts from linear algebra. Some applications in algebra, combinatorics, combinatorial geometry and design theory are illustrated by examples. Several well-known statements on the properties of the rank of a matrix, dimensions of subspaces of vector spaces, the criterion of the existence of a non-trivial solution of a homogeneous system of linear equations and the Cauchy-Schwarz inequality were used in the proofs. A generalized form of the Binet-Cauchy theorem on the determinant of the product of matrices is also included, as it is useful in applications. Among the results proved in the paper are Fisher’s inequality for combinatorial designs and the upper limit of the size of the equiangular set of directions in the d-dimensional Euclidean space. The latter result belongs to the topics in combinatorial geometry that have been investigated by mathematicians for decades and about which some significant results have been achieved in recent years