Let A be an n ×n symmetric matrix with real entries ai j, and let x→ be an n×1 column vector of the form x→ =x1⋮xn. Therefore, Qx1, … ,xn = x→T⋅A⋅x→ = ∑i,j=1nai j xi xj is said to be the quadratic form of A.
The expansion of Qx1, … , xn = x→T⋅A⋅x→
Qx1, ... , xn = x→T⋅A⋅x→
= x1 … xn⋅ a11…a1 n⋮an 1…an n⋅x1⋮xn
= x1 … xn⋅ ∑ a1 i⁢⋅xi⋮∑an i⁢⋅xi
= a11⋅x12 + a12⋅x1⋅x2 + ... + a1 n⋅x1⋅xn + a21⋅x2⋅x1 + a22⋅x22 + ... + a2 n⋅x2⋅xn + ... ... ... + an1⋅xn⋅x1 + an 2⋅xn⋅x2 + an n⋅xn2
A quadratic form, Q, and its corresponding symmetric matrix, A, can be classified as follows:
Positive definite if Q > 0 for all x→ ≠0.
Positive semi-definite if Q ≥0 for all x→ and Q = 0 for some x→ ≠0.
Negative definite if Q < 0 for all x→≠0.
Negative semi-definite if Q≤0 for all x→ and Q=0 for some x→≠0.
Indefinite if Q>0 for some x→ and Q<0 for some other x→.
If x→ has only two elements, x→ = x1x2 =xy, then we can graphically represent the quadratic form, Qx,y, as a function ℝ2 → ℝ. This is shown in the plot below.
This also allows us to visually determine the classification of the 2×2 symmetric matrix A as:
Positive definite if Qx,y is bounded below by z=0 and intersects this plane at only a single point, 0,0,0.
Positive semi-definite if Qx,y is bounded below by z = 0 and intersects this plane along a straight line.
Negative definite if Qx,y is bounded above by z=0 and intersects this plane at only a single point, 0,0,0.
Negative semi-definite if Qx,y is bounded above by z = 0 and intersects this plane along a straight line.
Indefinite if Qx,y lies above z=0 for some values of x→ and below z=0 for other values of x→, thereby intersecting this plane along a curve which is not a straight line.
Application in Multivariable Calculus
Using quadratic forms to classify matrices as definite, semi-definite, or indefinite can be useful in performing the multivariable second derivative test.
Let f: ℝ2 → ℝ have continuous second partial derivatives in some neighborhood of a critical point a,b and let Ha,b = fxx(a,b) fxy(a,b)fyx(a,b) fyy(a,b) be the Hessian matrix of f evaluated at a,b.
If Ha,b is positive definite, then a,b is a local minimum.
If Ha,b is negative definite, then a,b is a local maximum.
If Ha,b is indefinite, then a,b is a saddle point.
If Ha,bis positive semi-definite or negative semi-definite, then the second derivative test is inconclusive as to the nature of the point a,b.
Change the values in the symmetric matrix, A, and observe how the plot and formula of its quadratic form, Qx,y, change in response. The 3-D plot below can be rotated for visual representation.
Try to find a 2 × 2 symmetric matrix of each type: positive definite, positive semi-definite, negative definite, negative semi-definite, and indefinite.
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