Positive and negative matrix

Lemma: Let A=(aij)A=(a_{ij}) and B=(bij)B=(b_{ij}) be two positive semi-definite matrices, i.e.
n∑i,j=1aijxixj≥0\sum_{i,j=1}^{n} a_{ij} x_{i}x_{j} \geq 0 for all x=(x1,…,xn)∈Rnx=(x_1,…,x_n) \in R^n, similarly for B.
n∑i,j=1aijbij≥0\sum_{i,j=1}^{n}a_{ij}b_{ij} \geq 0.

Based on above lemma I want to show that: if A=(aij)A=(a_{ij}) is a symmetric positive matrix (A≥0A \geq 0) and B=(bij)B=(b_{ij}) is non-positive symetric matrix (B≤0B \leq 0) then n∑i,j=1aijbij≤0\sum_{i,j=1}^{n}a_{ij}b_{ij} \leq 0.

Thank you very much for help.




(−B)(-B) is positive definite. Can you finish from here?
– Alex R.
2 days ago



Yes, using lemma for A and -B gives −∑aijbij≥0-\sum a_{ij} b_{ij} \geq 0 thus ∑aijbij≤0\sum a_{ij} b_{ij} \leq 0. Thanks a lot, i dont know why I didn’t solve this in that way 😀
– Math seeker
2 days ago