Is strict/weak negative/positive definiteness/semidefiniteness of matrices preserved under matrix addition?
I tried to do this for 2x2 matrix but even this wasn't easy. (I tried to use the principal minors definition of definiteness)
2026-03-29 16:55:52.1774803352
Matrix addition and definiteness
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This is obvious if you use the definition that a matrix $A$ is positive definite if $x^TAx>0$ for all $x\ne 0$. Then we have $x^T(A+B)x=x^TAx+x^TBx>0$. The other kinds of definiteness are similar.