I have the following matrix $A$:

  • symmetric
  • all positive and/or zero values
  • the main diagonal is all the same value, $x$.

To ensure that the matrix $A$, is positive semidefinite, must I only ensure that $x \geq 0$? It seems correct from my thinking, but wanted to make sure. Thanks.


It is not sufficient to have positive diagonal entries. To see this, consider the matrix $$ A=\pmatrix{1& 10\\10& 1}. $$ It has the negative eigenvalue $-9$ to the eigenvector $$ v=\pmatrix{1\\-1}, $$and is thus not positive semi-definite.


Note however that a diagonally dominant symmetric matrix is positive semi definite.

In your case it is sufficient to add the condition $x \ge \sum_{j\neq i} A_{ij}, i\neq j$ to ensure $A$ to be positive semi definite.

  • $\begingroup$ Why does it suffice for $x\ge A_{ij}$? I would appreciate a reference for this if you have one. $\endgroup$ Feb 28 '18 at 1:25
  • $\begingroup$ @user3281410 The wikipedia article states this property for Hermitian diagonally dominant matrices, so it applies for real symmetric matrices as well : en.wikipedia.org/wiki/Diagonally_dominant_matrix $\endgroup$
    – Toool
    Mar 17 '18 at 15:06
  • $\begingroup$ I believe you mean to say $x\ge \sum_{j\neq i}^{}A_{ij}$ for all $i$, not just $x\ge A_{ij}$ for all $i\neq j$. There are symmetric, matrices with nonnegative entries such that the entries on the diagonal are all the same (denoted $x$) and satisfy $x \ge A_{ij}$ but $A$ is not positive semidefinite. See the first answer here: math.stackexchange.com/questions/2669941/… $\endgroup$ Mar 20 '18 at 22:56
  • $\begingroup$ @user3281410 Yep sorry I really meant $x \ge \sum_j A_{ij}, i\neq j$. I've updated the answer ! $\endgroup$
    – Toool
    Mar 21 '18 at 14:36
  • $\begingroup$ @Tool is the converse statement true? If I have a positive semidefinite matrix, can I say that it must also be diagonally dominant with nonnegative real entries on the diagonal? $\endgroup$
    – Tyberius
    Mar 28 '18 at 20:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.