# positive definite matrix and double non-negative matrix with 0-1 entries

If we have a positive definite (strictly) matrix $A$ and $M$ semi-positive definite with entries $0$ or $1$ and diagonal all ones,

What are the conditions to have $\operatorname{max eigenvalue}(A \circ M) < \operatorname{max eigenvalue}(A)$ ?

Here $\circ$ indicates the Hadamard product.

If $M$ is strictly positive definite, the condition is that $e_i$ is not an eigenvector of $A$, where $i$ indicates the index of the maximum diagonal element of $A$.

• Welcome to Math.SE! I edited your post to improve formatting (in particular using LaTeX for formulas). // I also have a question: what does "double" mean in the title of your post? – user53153 Feb 25 '13 at 5:12
• It means that is semi-positive definite and the entries are non-negative... – Liliana Forzani Feb 25 '13 at 14:42