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Quadratic Forms

Main Concept

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 =x1xn. Therefore, Qx1,  ,xn = xTAx = i,j=1nai j xi xj  is said to be the quadratic form of A.

The expansion of Qx1,  , xn = xTAx

Qx1, ... , xn = xTAx

        = x1  xn a11a1 nan 1an nx1xn

                   = x1  xn   a1 ixian ixi

        = a11x12 + a12x1x2 + ... + a1 nx1xn + a21x2x1 + a22x22 + ... + a2 nx2xn + ... ... ... + an1xnx1 + an 2xnx2 + an nxn2

        =i,j=1nai jxixj

 

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 x0.

• 

Negative semi-definite if Q0 for all x and Q&equals;0 for some x0.

• 

Indefinite if Q&gt;0 for some x and Q<0 for some other x.

 

Graphical Representation

If x has only two elements, x &equals; x1x2 &equals;xy, then we can graphically represent the quadratic form, Qx&comma;y, as a function 2  .  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&comma;y is bounded below by z&equals;0 and intersects this plane at only a single point, 0&comma;0&comma;0&period;

• 

Positive semi-definite if Qx&comma;y is bounded below by z &equals; 0 and intersects this plane along a straight line.

• 

Negative definite if Qx&comma;y is bounded above by z&equals;0 and intersects this plane at only a single point, 0&comma;0&comma;0&period;

• 

Negative semi-definite if Qx&comma;y is bounded above by z &equals; 0 and intersects this plane along a straight line.

• 

Indefinite if Qx&comma;y lies above z&equals;0 for some values of x and below z&equals;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&colon; &reals;2  &reals; have continuous second partial derivatives in some neighborhood of a critical point a&comma;b and let Ha&comma;b &equals; fxx&lpar;a&comma;b&rpar;     fxy&lpar;a&comma;b&rpar;fyx&lpar;a&comma;b&rpar;     fyy&lpar;a&comma;b&rpar; be the Hessian matrix of f evaluated at a&comma;b.

• 

If Ha&comma;b is positive definite, then a&comma;b is a local minimum.

• 

If Ha&comma;b is negative definite, then a&comma;b is a local maximum.

• 

If Ha&comma;b is indefinite, then a&comma;b is a saddle point.

• 

If Ha&comma;bis positive semi-definite or negative semi-definite, then the second derivative test is inconclusive as to the nature of the point a&comma;b&period;

 

Change the values in the symmetric matrix, A, and observe how the plot and formula of its quadratic form, Qx&comma;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.

xT

A

x

&equals;

xTAx

xy

 

xy

&equals;

 

 

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