Naslov
A remark on Hölder's inequality for commuting matrices,

Autori
Mond, Bertram ; Pečarić, Josip ;

Izvornik
Southeast Asian bulletin of mathematics (0218-0006) 20 (1996), 1; 19-21

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
Hermitian matrix; positive definite matrix; eigenvalue

Sažetak
Let $A$ and $B$ be commuting $n imes n$ matrices over the field of complex numbers which are Hermitian and positive definite. Let $lambda_1 geq lambda_2 geq ldots lambda_n$ be the eigenvalues of $A$ each appearing as many times as its multiplicity, and, similary, let $mu_1geq mu_2 geq ldots mu_n$ be the eigenvalues of $B$. Assume that $lambda_1mu_1>lambda_nmu_n$. Let $p>1$ and $q$ be real numbers with $1/p+1/q=1$. Set $gamma=(lambda_1/lambda_n)^{1/q}(mu_1/mu_n)^{1/p}$. Then for any non-zero $n$-dimensional vector $x$ holds $$ 1leq frac{(A^px,x)^{1/p}(B^qx,x)^{1/q}}{(Ax,Bx)}leq (frac{q}{p+q} frac{gamma^p-gamma^{-q}}{1-gamma^{-q}})^{1/p} (frac{p}{p+q} frac{gamma^p-gamma^{-q}}{gamma^{p}-1})^{1/q}.$$

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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