1 Definition and notation

A principal minor of a square matrix is the determinant of a submatrix obtained by choosing a set of indices and using the same set for both rows and columns. This construction preserves a structural “principal” relationship between the chosen coordinates and the original matrix.

1.1 Square submatrices

Let \(A\in \mathbb{F}^{n\times n}\) and let \(\alpha=\{i_1,\dots,i_k\}\subseteq \{1,\dots,n\}\) with \(\alpha=k\). The principal submatrix associated with \(\alpha\) is

\[ A[\alpha]=A[\alpha,\alpha]\in \mathbb{F}^{k\times k}, \] meaning that its \((r,s)\)-entry equals \(A_{i_r,i_s}\). The principal minor corresponding to \(\alpha\) is then \[ \det(A[\alpha]). \] By varying \(\alpha\), one obtains principal minors of all sizes \(k=1,\dots,n\), including the full determinant when \(k=n\).

1.2 Principal index sets

The index set \(\alpha\) is usually referred to as a principal index set. Common conventions include:

  • treating \(\alpha\) as an unordered set, with the determinant being unaffected by how the indices are listed (up to sign, which cancels for determinants under consistent reindexing), and
  • denoting the size by \(k\), so that principal minors are often grouped by order.

For symmetric or Hermitian matrices, principal minors play a particularly prominent role because they encode local information about quadratic forms restricted to coordinate subspaces.

1.3 Principal minors versus ordinary minors

An ordinary minor is the determinant of any square submatrix formed by selecting arbitrary row indices and arbitrary column indices, not necessarily matching. In contrast:

  • a principal minor uses the *same* index set for rows and columns, i.e., it is of the form \(\det(A[\alpha,\alpha])\).

This restriction is not cosmetic: many deep results (especially definiteness criteria) rely on the principal structure and fail if one allows non-matching row/column choices.

2 Types of principal minors

Principal minors are commonly categorized by how the index sets are selected.

2.1 Leading principal minors

The leading principal minors of \(A\in \mathbb{F}^{n\times n}\) are determinants of the top-left \(k\times k\) blocks: \[ \Delta_k=\det(A[ \{1,2,\dots,k\} ])\quad (k=1,\dots,n). \] They form a sequence \(\Delta_1,\Delta_2,\dots,\Delta_n\) aligned with nested coordinate subspaces \(\mathrm{span}(e_1,\dots,e_k)\).

2.2 Non-leading principal minors

Any principal minor that is not leading is termed non-leading. For example, \(\det(A[\{1,3\},\{1,3\}])\) is principal but not leading in dimensions \(n\ge 3\). Non-leading minors appear in refined criteria where leading minors alone do not capture enough information.

2.3 Proper principal minors

A proper principal minor is a principal minor of order strictly less than \(n\). Thus, it excludes \(\det(A)\) (the order-\(n\) principal minor). Proper principal minors are often the ones examined when studying properties like definiteness or rank conditions via submatrices.

3 Basic properties

Principal minors inherit several structural features from the determinant and from the way index sets are chosen.

3.1 Determinantal interpretation

For a principal index set \(\alpha\), the principal minor \[ \det(A[\alpha]) \] is the determinant of the restriction of \(A\) to the coordinate subspace spanned by \(\{e_i:i\in\alpha\}\). In the symmetric/Hermitian setting, this restriction corresponds to evaluating a quadratic form on that subspace, which explains why sign patterns of principal minors can determine definiteness.

3.2 Symmetry under permutation of indices

If \(P\) is a permutation matrix, then \(PAP^T\) corresponds to simultaneously permuting rows and columns of \(A\). Principal minors of \(PAP^T\) match those of \(A\) up to relabeling of index sets: \[ \det\big((PAP^T)[\alpha]\big)=\det\big(A[\pi(\alpha)]\big), \] where \(\pi\) is the induced permutation on indices. Thus, the collection of all principal minors is invariant under simultaneous permutation of coordinates (though the leading minors depend on the chosen ordering).

3.3 Relation to matrix size

A matrix of size \(n\) has principal minors of orders \(1\) through \(n\). The number of principal minors of a fixed order \(k\) equals \(\binom{n}{k}\), since each choice corresponds to an index set \(\alpha\) of size \(k\). This combinatorial growth is a reason computations often focus on leading or selected principal minors in practice.

4 Computation

While principal minors are defined by determinants, practical computation can use several strategies.

4.1 Direct determinant evaluation

The most straightforward method computes \(\det(A[\alpha])\) for each index set \(\alpha\) of interest using standard determinant algorithms (Gaussian elimination, LU factorization, or direct formulas for small sizes). This is typically feasible when only a small number of principal minors are needed.

4.2 Expansion by index subsets

For symbolic manipulation or when relating minors to characteristic polynomials, one can employ determinant identities that express minors through sums over index subsets. These expansions are useful when:

  • principal minors are linked to coefficients of polynomials, or
  • one wants to compute families of minors efficiently for structured matrices.

In general, subset expansions can become computationally expensive because the number of index sets grows combinatorially.

4.3 Recursive methods

When determinants of many principal submatrices are required, recursive approaches can reuse intermediate computations. Examples include:

  • dynamic programming over nested index sets (especially for leading minors), and
  • incremental updates for structured matrices where elimination steps share common pivots.

Such methods reduce repeated work, although the feasibility depends strongly on sparsity, symmetry, and the size/order of minors being targeted.

5 Key theorems involving principal minors

Several central theorems use principal minors as tests or as bridges to other matrix invariants.

5.1 Sylvester’s criterion

For a real symmetric (or complex Hermitian) matrix \(A\), Sylvester’s criterion characterizes definiteness using principal minors.

5.1.1 Positive definite matrices

\(A\) is positive definite if and only if all leading principal minors are positive: \[ A \succ 0 \quad \Longleftrightarrow \quad \Delta_k=\det(A[\{1,\dots,k\}])>0 \ \text{for all }k=1,\dots,n. \] Equivalently, the matrix has a factorization consistent with positivity (e.g., Cholesky) that forces these determinants to have the required sign pattern.

5.1.2 Negative definite matrices

Similarly, \(A\) is negative definite if and only if the leading principal minors alternate signs with order: \[ A \prec 0 \quad \Longleftrightarrow \quad (-1)^k\,\Delta_k>0 \ \text{for all }k. \] This again reflects the relationship between sign changes in determinants and the induced definiteness of the associated quadratic form.

5.2 Jacobi’s theorem

Jacobi’s theorem relates minors of a matrix to minors of its inverse (when the inverse exists). In a common formulation, principal minors of \(A^{-1}\) correspond to complementary principal minors of \(A\). Conceptually, it expresses that determinants of sub-blocks behave compatibly under inversion, with complementary index sets playing the role of “dual” information.

5.3 Principal minor assignment problem

The principal minor assignment problem asks whether a prescribed family of numbers \(\{\mu_\alpha\}\), indexed by all principal index sets \(\alpha\), can arise as principal minors of some matrix (often with additional constraints such as symmetry). This problem is studied in different levels of generality:

  • determining existence,
  • characterizing uniqueness,
  • and describing which algebraic consistency conditions must hold.

Because principal minors are highly structured determinants, not every arbitrary assignment is realizable; the constraints come from determinant relations among overlapping submatrices.

6 Special cases

Certain matrix classes make principal minors especially transparent.

6.1 1×1 and 2×2 matrices

For \(1\times 1\) matrices \(A=[a]\), the only principal minor is \(a\), equal to the determinant.

For \(2\times 2\) matrices \[ A=\begin{pmatrix} a&b\\ c&d\end{pmatrix}, \] the principal minors are:

  • order 1: \(a\) and \(d\),
  • order 2: \(\det(A)=ad-bc\).

If \(A\) is symmetric (\(c=b\)), these minors are directly tied to definiteness tests.

6.2 Diagonal matrices

If \(A=\mathrm{diag}(\lambda_1,\dots,\lambda_n)\), then for an index set \(\alpha\), \[ \det(A[\alpha])=\prod_{i\in\alpha}\lambda_i. \] Thus, principal minors factor neatly into products of diagonal entries, making sign and rank behavior immediate.

6.3 Triangular matrices

For an upper or lower triangular matrix, any principal submatrix is also triangular, so \[ \det(A[\alpha])=\prod_{i\in\alpha} a_{ii}, \] where \(a_{ii}\) are the diagonal entries of \(A\). Off-diagonal entries do not affect principal minor values in this case.

7 Applications

Principal minors appear throughout matrix theory as diagnostic quantities.

7.1 Matrix definiteness tests

The most prominent application is definiteness testing for symmetric/Hermitian matrices. Sylvester’s criterion uses leading principal minors, while related criteria can involve broader families of principal minors when leading ones are insufficient or when pivoting/order matters.

7.2 Rank and invertibility

Principal minors encode whether a submatrix is singular. For a given \(\alpha\), \(\det(A[\alpha])=0\) indicates that the restriction to the corresponding coordinate subspace fails to be invertible. In particular:

  • \(A\) is invertible exactly when the full principal minor \(\det(A)\neq 0\).
  • More generally, vanishing of all principal minors of a certain order signals limitations on rank growth across coordinate restrictions.

7.3 Spectral theory

Principal minors connect to spectral information via characteristic polynomials. For example, coefficients of \(\det(tI-A)\) can be expressed in terms of sums of principal minors of \(A\), tying eigenvalue data to determinant expansions. Through this link, principal minor computations can provide algebraic access to eigenvalue-related invariants without diagonalization.

7.4 Combinatorial matrix theory

In combinatorial settings, determinant expressions involving minors often correspond to weighted counts of combinatorial objects (such as spanning trees and matchings) depending on the matrix model (e.g., Laplacians, adjacency-derived constructions). Principal minors arise naturally because they correspond to determinants of induced substructures, making them central in connecting linear algebra with discrete enumeration.

Principal minors are part of a broader network of determinant-based matrix concepts.

8.1 Principal submatrices

A principal submatrix is the submatrix \(A[\alpha]\) itself, before taking determinants. Many results are stated equivalently for principal submatrices (in terms of invertibility, inertia, or restrictions of quadratic forms), with determinants then translating those properties into scalar invariants.

8.2 Cofactors and adjugates

Cofactors and the adjugate \(\operatorname{adj}(A)\) are built from determinants of minors, typically with sign patterns. While cofactor minors need not be principal, they share a conceptual basis: determinant minors arranged according to specific deletion rules. In computations involving \(A^{-1}=\operatorname{adj}(A)/\det(A)\), principal minors indirectly influence determinant expansions and sensitivity to singularities.

8.3 Characteristic polynomial coefficients

For \(A\in\mathbb{F}^{n\times n}\), the characteristic polynomial \[ \chi_A(t)=\det(tI-A) \] has coefficients that can be described using sums of principal minors of \(A\). This makes principal minors a bridge between local determinant data and global eigenvalue information, allowing one to express spectral invariants without direct computation of eigenvalues.