# Is it true: A real symmetric matrix is either positive definite or negative definite or indefinite?

I got a real symmetric matrix that is neither positive definite or negative definite, so can I just say that this matrix is indefinite?

• Is the zero matrix positive/negative definite or indefinite? – Santiago Canez Sep 18 '14 at 4:41
• Thanks Santiago! Then how about this definition: a real symmetric matrix H is indefinite if x'Hx is positive for some nonzero real vector x and is negative for some nonzero real vector x. Is this definition correct? – larrybr Sep 18 '14 at 17:50

No. You have described all the real symmetric matrices with nonzero determinant. The others are usually called semidefinite, for example $$\left( \begin{array}{rr} 1 & 0 \\ 0 & 0 \end{array} \right)$$

It is enough to consider diagonal matrices here, because real symmetric matrices can be orthogonally diagonalized. There are three counts; first the matrix will be $n$ by $n.$ We call the number of positive diagonal entries $n_+,$ the number of negative diagonal entries $n_-,$ then the number of zero diagonal entries $n_0.$ As these make up the entire diagonal we have $$n_+ + n_0 + n_- = n$$

If all three, $n_, n_0 , n_-,$ are nonzero, I would probably say that the form is indefinite but add that it is "degenerate," by which I mean the rank is less than $n,$ the determinant is nonzero and so on.

• Thanks Will! Do you think my definition above in the comment is correct? – larrybr Sep 18 '14 at 17:54
• @larrybr, why do you want a definition? – Will Jagy Sep 18 '14 at 18:08
• Hmm, your opening paragraph is a bit inaccurate. "The others" (those real symmetric matrices with zero determinants) can be indefinite unless $n\le2$. – user1551 Sep 19 '14 at 23:51

Yes.

By the way, check out Wolfram's definition of Indefinite Matrix

http://mathworld.wolfram.com/IndefiniteMatrix.html

• This is not correct. See my comment above. – Santiago Canez Sep 18 '14 at 4:41
• You're right Santiago. Then any given square matrix has exactly one of the following labels: positive definite, negative definite, indefinite, the zero matrix. – Romeo Sep 18 '14 at 4:48
• No, the matrix given in Will Jagy's answer is none of those. You're missing the "semidefinite" possibilities. – Santiago Canez Sep 18 '14 at 16:17