The variance matrix must be positive semidefinite

Problem

\(X\) is an \(n×1\) random vector, \(X=\left( X_1, \ldots ,X_n \right)\) . Explain why the variance matrix \(Var\left( X \right)\) must be positive semidefinite.

Hint: Let \(a\) be an arbitrary \(n×1\) vector of constants and \(Y=a'X\) . What can we say about \(Var\left( Y \right)\) ?

Solution

\(Y\) is a random variable and \(Var\left( Y \right)≥0\) must hold no matter which \(a\) we pick. Since

\[Var\left( Y \right)=a'Var\left( X \right)a\]

we see that \(a'Var\left( X \right)a≥0\) for all choices of \(a\) which is the same as saying that \(Var\left( X \right)\) must be positive semdefinite.

  • Random variable: \(Var\left( X \right)≥0\)
  • Random vector: \(Var\left( X \right)\) is positive semidefinite
  • Note that if \(Var\left( X \right)\) is positive definite then \(Var\left( Y \right)>0\) for all choices of \(a\) except \(a=0\) .