79. Principal Minor

A principal minor of an $𝑛\times 𝑛$ matrix $𝐴$ is the determinant of a square submatrix whose diagonal is a subset of $𝐴$’s diagonal.

To form a principal minor: pick a subset of indices $𝐼\subseteq \left\{1,2,\dots ,𝑛\right\}$, then delete every row and column not in $𝐼$. The resulting submatrix is square; its determinant is the principal minor.

Pick rows/columns $𝐼=\left\{1,3\right\}$:

79.1. Levels (order)

A principal minor formed from $|𝐼|=𝑘$ indices is called a level-$𝑘$ (or order-$𝑘$) principal minor.

For an $𝑛\times 𝑛$ matrix:

• There are $𝑛$ levels (from $𝑘=1$ to $𝑘=𝑛$)
• The number of level-$𝑘$ principal minors is $\left(\genfrac{}{}{0ex}{}{𝑛}{𝑘}\right)$
• Total number of principal minors: ${\sum }_{𝑘=0}^{𝑛}\left(\genfrac{}{}{0ex}{}{𝑛}{𝑘}\right)=2^{𝑛}$ (including the trivial empty one)

79.2. Where they show up

• Sylvester’s criterion (positive-definiteness test): a symmetric matrix is positive definite iff every leading principal minor is positive
• Positive semi-definiteness: requires every principal minor (not just leading) to be non-negative
• Convex analysis — checking definiteness of the Hessian via principal minors

79.3. See also

• Leading Principal Minor — minors from the top-left corner
• Minor — general minors (rows and columns can differ)
• Determinant