Leading Principal Minor

80. Leading Principal Minor

A leading principal minor of order $𝑘$ is the determinant of the top-left $𝑘 \times 𝑘$ submatrix of $𝐴$ .

That is, take rows and columns indexed $1, 2, \dots, 𝑘$ — no other choices.

For a $3 \times 3$ matrix, the three leading principal minors are:

Order 1: $𝑎_{11}$ . Order 2: $det [\begin{matrix} 𝑎_{11} & 𝑎_{12} \\ 𝑎_{21} & 𝑎_{22} \end{matrix}]$ . Order 3: $det (𝐴)$ itself.

A symmetric matrix $𝐴$ is positive definite iff all leading principal minors are strictly positive:

Δ_{1} > 0, Δ_{2} > 0, \dots, Δ_{𝑛} > 0

where $Δ_{𝑘}$ is the order- $𝑘$ leading principal minor.

For negative definite: signs alternate — $Δ_{1} < 0, Δ_{2} > 0, Δ_{3} < 0, \dots$ (i.e. ${(- 1)}^{𝑘} Δ_{𝑘} > 0$ ).

Example

𝐴 = [\begin{matrix} 2 & - 1 \\ - 1 & 3 \end{matrix}]

$Δ_{1} = 2 > 0$ , $Δ_{2} = 2 \cdot 3 - (- 1) (- 1) = 5 > 0$ . So $𝐴$ is positive definite.