[math-fun] Positive eigenvalues imply positive definite?

I found this book review in the April 2020 Physics Today [ https://physicstoday.scitation.org/doi/10.1063/PT.3.4456 ]. The following is a quote from it. "After stating that “an arbitrary non-singular tensor T is positive definite if v·T·v>0 for all vectors v≠0,” the book goes on to spread the myth that positive eigenvalues of T are sufficient for T to be positive definite; the 2 × 2 matrix T={{4,9},{1,4}} with v={1,−1} is a counterexample." Then in reading Arthur Gelb, "Applied Optimal Estimation", I found problem 2-3, in which A is a matrix. "Show that A is positive definite if and only if all of its eigenvalues are positive." -- Gene