What is a principal minor of a matrix?

Question

I was going through the book on operation research by Hamdy A.Taha. It referred to principal minor of a hessian matrix. Can someone explain what is meant by a principal minor? Is it different from 'minor of a matrix'?

Answer 1

As pointed out by @RobertIsrael, the principal minor is a minor in which the indices of the omitted row and column match. for example for a $3*3$ matrix: a principal minor can be created by omitting '1st row and 1st column', or by omitting '1st row, 2nd row, 1 column, 2nd column' and so on.

Answer 2

A minor of a matrix $A$ is the determinant of a submatrix formed from $A$ by deleting some (possibly empty,) set of rows and some (possibly empty) set of columns. In order for this determinant to exist, the number of remaining rows must be equal to the number of remaining columns (and there must be at least one remaining row and column). A principal minor of a square matrix is one where the indices of the deleted rows are the same as the indices of the deleted columns.

Thus for a $3 \times 3$ matrix $A$, you could delete nothing (resulting in the determinant of the matrix itself), delete one row and the corresponding column (resulting in one of three possible $2 \times 2$ determinants), or delete two rows and the corresponding two columns (resulting in one of the three diagonal elements).

Answer 3

Quoting from here:

A principal minor is simply the determinant of a submatrix obtained from A when the same set of rows and columns are stricken out.

What is a minor of a matrix?

Let's deal with a square matrix $A \in \mathbb{R}^{N \times N}$ for the moment. A minor of a square matrix is a real number; it is the determinant obtained by "striking out" or "deleting" certain rows & columns of a matrix.

Concretely, let $R = \{R_1, ..., R_m\}$ be the set of $m$ rows we want to delete. So, if we want to delete rows 1, 3, 4, we have $R = \{R_1=1, R_2=3, R_3=4\}$. Similarly, let $C = \{C_1, ..., C_m\}$ be the set of $m$ columns we want to delete.

Because we start with a square matrix and want to calculate a determinant, we must also end up with a square matrix. Thus, we must delete an equal number of rows and columns, i.e. $|R| = |C|$. This is a basic requirement to obtain minors of a matrix; the number of rows & columns deleted must be the same. Let's continue to call this number $m$.

Other than this requirement, the sets $R$ and $C$ need not be the same; we could have $R = \{1, 3, 4\}$ and $C = \{2, 4, 5\}$. We can still calculate the determinant of the remaining matrix (which will be a square matrix of dimensions $(N-m) \times (N-m) = 2 \times 2$).

As a concrete example, suppose we have $A \in \mathbb{R}^{5 \times 5}$. We have a few options of $R$ and $C$:

• If we pick $m=4$, we remove any 4 rows & columns, leaving only a single element. Thus the minor becomes the determinant of that element (which is the element itself). This is called a first-order minor. There are $5C4 \times 5C4 = 5 \times 5 = 25$ such first-order minors in a $5 \times 5$ matrix (each element is a first-order minor).
• If we pick $m=3$, we remove any 3 rows & columns, giving us a 2x2 submatrix. The determinant of this matrix (which is calculated in the usual way) is called the second-order minor. There are $5C3 \times 5C3 = 10 \times 10 = 100$ such minors.
• If we pick $m=3$, we remove any 2 rows & columns, giving us a 3x3 submatrix. This is where start to calculate the determinant recursively from the resulting submatrix using the order-2 minors and the cofactors $c_{ij}$, multiplying each cofactor by $(-1)^{(i+j)}$, etc. There are $5C2 \times 5C2 = 10 \times 10 = 100$ such minors.
• Continuing, a k-th order minor is the determinant of any $k \times k$ submatrix of $A \in \mathbb{R}^{N \times N}$. There are ${}^NC_{(N-k)} \times {}^NC_{(N-k)} = {}^NC_{k} \times {}^NC_{k}$ such k-th order minors.

What are Principal minors?

So far, we have allowed $R$ and $C$ to be different sets, under the requirement that $|R| = |C|$. If we add a further constraint that $R = C$, i.e. we delete the same indexes of rows and columns, then we get a bunch of "principal" minors. Since we pick the same rows as columns, we only have ${}^NC_{(N-k)} = {}^NC_k$ such "principal minors".