7.2 - Quadratic Forms quadratic form on is a function defined on whose value at a vector in can be computed by an expression of the form, where is an symmetric matrix. The matrix R n Q R n x R n Q(x) = is called the matrix of the quadratic form. x T x n n x 1 Let. Compute x x for the following matrices: [ x 2 ] T a) b) 4 0 = [ 0 3 ] 3 2 = [ 2 7 ] x R 3 Q(x) = 5 x1 2 + 3 x2 2 + 2x3 x 1 x 2 + 8x 2 x 3 x T x x1 2, 2, 3 x i x j i j (i, j) (j, i). The coefficient of x 1 x 3 is 0.) For in, let. Write the quadratic form as. (Note: the coefficients of go on the diagonal of. To make symmetric, the coefficients of for must be split evenly between the - and -entries in Q(x) = x1 2 8x 1 x 2 5x2 Q(x) 3 2 x =, [ 1 ] [ 2 ] 1. [ 3 ] Let. Compute the value of for and In some cases, quadratic forms are easier to use when they have no cross-product terms - - that is, when the matrix of the quadratic form is a diagonal matrix. Fortunately, the crossproduct term can be eliminated by making a suitable change of variable. Change of Variable in a Quadratic Form If x represents a variable vector in R n, the a change of variable is an equation of the form or equivalently x = Py y = P 1 x
P y R n y x R n P where is an invertible matrix and is a new variable vector in. Here is the coordinate vector of relative to the basis of determined by the columns of. If the change of variable is made in a quadratic form, then and the new matrix of the quadratic form is P T P. Since is symmetric, Theorem 2 guarantees that there is an orthogonal matrix P such that P T P is a diagonal matrix D, and the quadratic form becomes. x T x x T x = (Py) T (Py) = y T P T Py = y T ( P T P)y y T Dy Make a change of variable that transforms the quadratic form into a quadratic form with no cross-product term. a) Find the matrix of the quadratic form. b) Diagonalize. Q(x) = x1 2 8x 1 x 2 5x2 2 x 1 y 1 c) Substitute x = Py, where x = and y = into the quadratic form. [ x 2 ] [ y 2 ] Theorem - The Principal xes Theorem Let be an symmetric matrix. Then there is an orthogonal change of variable, x = Py, that transforms the quadratic form x T x into a quadratic form y T Dy with no cross-product term. The columns of P in the theorem are called the principal axes of the quadratic form x T x. The vector is the coordinate vector of relative to the orthonormal basis of given by these principal axes. Geometric View of Principal xes If, where is an invertible symmetric matrix, and if is a constant, then Q(x) = n n corresponds to an ellipse, a hyperbola, two intersecting lines or a single point or contains no points at all. If y x R n x T x 2 2 c x T X = c is a diagonal matrix, the graph is in standard position, as shown below.
If is not a diagonal matrix, the graph is rotated out of standard position, as shown below. Finding the principal axes (determined by the eigenvectors of ) amounts to finding a new coordiante system with respect to which the graph is in standard position. The rotated hyperbola shown above is the graph of the equation x T x = 16, where is that matrix above. The positive y 1 -axis in the rotated hyperbola is in the direction of the first column of the matrix, above, and the positive -axis is in the direction of the P second column of. P y 2 The rotated ellipse, above, is the graph of the equation 5x1 2 4 x 1 x 2 + 5 x2 2 = 48 change of variable that removes the cross-product term from the equation.. Find a
Classifying Quadratic Forms When is an n n matrix, the quadratic form Q(x) = x T x is a real-valued function with R n R 2 x = ( x 1, x 2 ) Q ( x 1, x 2, z) z = Q(x) x = 0 Q(x) domain. The figure below displays the graphs of four quadratic forms with domain. For each point in the domain of a quadratic form, the graph displays the point where. Notice that except at, the values of are all positive in the first figure and all negative in the last. The horizontal cross-sections of the first and last graphs are ellipses, and hyperbolas for the second and third graphs. Definition quadratic form is: positive definite if for all. negative definite if for all. indefinite if assumes both positive and negative values. lso, is said to be positive semidefinite if for all, and to be negative semidefinite if Q Q(x) Q(x) > 0 x 0 Q(x) < 0 x 0 Q Q(x) 0 x Q(x) 0 x for all. The quadratic forms in the first two figures above are both positive semidefinite, but the first form is better described as positive definite.
Theorem 5 - Quadratic Forms and Eigenvalues Let be an n n symmetric matrix. Then a quadratic form x T x is: positive definite if and only if the eigenvalues of negative definite if and only if the eigenvalues of indefinite if and only if are all positive. are all negative. has both positive and negative eigenvalues. Is Q(x) = 3 x1 2 + 2 x2 2 + x3 2 + 4 x 1 x 2 + 4x 2 x 3 positive definite? The classification of a quadratic form is often carried over to the matrix of the form. Thus a positive definite matrix is a symmetric matrix for which the quadratic form x T x definite. Other terms, such as positive semidefinite matrix are defined analogously. is positive