I know that pd (positive definite) matrices are dense in psd matrices. Let $A$ be TPSD (tridiagonal positive semi definite) with all entries positive. Does there exist a sequence of TPD matrices with all entries positive, which converge to $A$ ? Any small tips or suggestion is really appreciated. Thanks
2026-03-25 05:07:04.1774415224
denseness of tridiagonal positive semidefinitness in tridiagonal positive definiteness
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Let $A$ be a $n\times n$ tridiagonal positive semi definite (there is a mistake in your post) with all entries positive. Then $B_k=A+\dfrac{1}{k}I_n$ is tridiagonal positive definite with all entries positive (for every positive integer $k$). Moreover $B_k$ tends to $A$ when $k$ tends to $\infty$.