Definite matrices

Summary

• \(A\) is a square \(n×n\) matrix and \(x\) is an \(n×1\) vector. We say that \(A\) is
• Positive definite if \(x'Ax>0\) for all \(x≠0\)
• Negative definite if \(x'Ax<0\) for all \(x≠0\)
• Positive semidefinite if \(x'Ax≥0\) for all \(x≠0\)
• Negative semidefinite if \(x'Ax≤0\) for all \(x≠0\)
• Indefinite: \(x'Ax\) may take values of different sign
• Result: If \(A\) is diagonal then \(A\) is
• Positive definite if \(A_{ii}>0\) for \(i=1, \ldots ,n\)
• Negative definite if \(A_{ii}<0\) for \(i=1, \ldots ,n\)
• Positive semidefinite if \(A_{ii}≥0\) for \(i=1, \ldots ,n\)
• Negative semidefinite if \(A_{ii}≤0\) for \(i=1, \ldots ,n\)
• Result: for \(n=2\) and

\[A=\begin{bmatrix}a & c \\ c & b\end{bmatrix} x=\begin{bmatrix}x_1 \\ x_2\end{bmatrix}\]

• we have

\[x'Ax=ax_1^2+bx_2^2+2cx_1x_2\]

• and \(A\) is
• Positive definite if \(\left| A \right|>0\) and \(a>0\)
• Negative definite if \(\left| A \right|>0\) and \(a<0\)
• Positive semidefinite if \(\left| A \right|≥0\) and \(a≥0\)
• Negative semidefinite if \(\left| A \right|≥0\) and \(a≤0\)
• where \(\left| A \right|\) is the Determinant of \(A\) .