We consider the characterizations of positive definite as well as nonnegative definite quadratic forms in terms of the principal minors of the associated symmetric matrix. We briefly review some of the known proofs, including a classical approach via the Lagrange-Beltrami identity. For quadratic forms in up to 3 variables, we give an elementary and self-contained proof of Sylvester's Criterion for positive definiteness as well as for nonnegative definiteness. In the process, we obtain an explicit version of Lagrange-Beltrami identity for ternary quadratic forms.
Let A = (a ij ) be an n × n real symmetric matrix and
a ij x i x j be the corresponding (real) quadratic form in n variables. Recall that the matrix A or the form Q is said to be positive definite (resp: nonnegative definite 1 ) if Q(x) > 0 (resp: Q(x) ≥ 0) for all x ∈ R n , x = 0. For any matrix, a minor is the determinant of a square submatrix. A minor is called a principal minor if the rows and columns chosen to form the submatrix have the same indices; further, if these indices are consecutive and start from 1, then it is called a leading principal minor. Sylvester’s minorant criterion is the well-known result that A is positive definite ⇐⇒ the leading principal minors of A are positive.
(
An analogous characterization for nonnegative definite matrices seems relatively less well-known and conspicuous by its absence in most texts on Linear Algebra. It may be tempting to guess that A is nonnegative definite if and only if all the leading principal minors of A are nonnegative. In fact, some books (e.g., [1, Ch.
2 §5] or [12, p. 133]) appear to state incorrectly that this is true. To see that nonnegativity of leading principal minors does not imply nonnegative definiteness, it suffices to consider the matrix
In [3, p.293], this example is given and the author also states that for positive semi-definiteness, there is no straightforward generalization of Sylvester’s minorant criterion! Nonetheless there is a simple and natural generalization as follows.
A is nonnegative definite ⇐⇒ the principal minors of A are nonnegative. (2) The aim of this article is to effectuate a greater awareness of (2) and a related algebraic fact known as the Lagrange-Beltrami identity. The existing proofs in the literature of (2) as well as (1) seem rather involved. (See Remark 2.2 and the paragraph before Proposition 2.1.) With this in view, we shall outline a completely self-contained and elementary proof of ( 1) and ( 2) when n ≤ 3. There is a good reason why such a proof may be useful and interesting. As is well-known, characterizations of positive definite matrices, when applied to the Hessian matrix, play a crucial role in the local analysis of real-valued functions of several real variables. For example, they give rise to the so called Discriminant Test, which is a useful criterion to determine a local extremum or a saddle point. Characterizations of nonnegative definiteness are also useful here, and more importantly, in the study of convexity and concavity of functions of several variables. (See, for example, [10, §42].) Usually these topics are studied in Calculus courses before the students have an exposure to Linear Algebra and learn notions such as eigenvalues and results such as the Spectral Theorem. Also, it is common to restrict to functions of two or three variables. Thus it seems desirable to have a proof for n ≤ 3 that assumes only the definition of the determinant of a 2 × 2 or 3 × 3 matrix. As indicated earlier, the pursuit of an elementary proof leads one to the Lagrange-Beltrami identity, which is yet another gem from classical linear algebra that deserves to be better known and better understood. In Section 2 below, we explain this identity and illustrate its use in proving (1) and (2) in the simplest case n = 2. We also comment on some of the existing proofs of (1) and (2) in the general case. Section 3 deals with the case n = 3, and ends with a number of remarks and a related question.
Let A = (a ij ) and Q(x) be as in the Introduction.
is a sum of squares that vanishes only when x = 0. The Lagrange-Beltrami identity does just this. It states that if the product
and each b ij is a rational function in the entries of A. Notice that the equations for y 1 , . . . , y n in terms of x 1 , . . . , x n are in a triangular form; hence if
To prove the other implication ‘⇒’ in (1), it is customary to use induction on n together with the following fact.
A is positive definite =⇒ det A > 0.
(
This fact follows readily from the following eigenvalue characterization:
A is positive definite ⇐⇒ the eigenvalues of A are positive.
In turn, (5) follows from the Spectral Theorem for real symmetric matrices. In the case of nonnegative definiteness, we have a straightforward analogue of (5), namely,
A is nonnegative definite ⇐⇒ the eigenvalues of A are nonnegative.
This, too, follows from the Spectral Theorem. Also, as a consequence, we have an obvious analogue of ( 4) that together with induction on n will prove the implication ‘⇒’ in (2). However, if one is seeking an elementary proof, one should try to avoid the use of the Spectral Theorem and its consequences such as ( 5) and ( 6). Also, if some ∆ i = 0, then to prove (2), the Lagrange-Beltrami identity (3) seems useless even if we clear the denominators. We will now see that at least for small values of n, the Lagrange-Beltrami identity is still useful if we know it explicitly and also if we know some of its avatars. Moreover, the use of Spectral Theorem ca
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