page 1 of Section 9.1 CHAPTER 9 QUADRATIC FORMS SECTION 9.1 THE MATRIX OF A QUADRATIC FORM quadratic forms and their matrix notation 2 2 2 Ifq=a1 x +a2 y +a3 z +a4 xy+a5 xz+a6 yz then q is called a quadratic form (in variables x,y,z). There is a q value (a scalar) at every point. (To a physicist, q is probably the energy of a system with ingredients x,y,z.) The matrix for q is 1 1 a a a 1 2 4 2 5 1 1 A= a a a 2 4 2 2 6 1 1 a a a 2 5 2 6 3 It's the symmetric matrix A with this connection to q: x (1) q = []xyz A y z or equivalently x Û T Û Û (2) q = x Ax where x = y z example 1 2 2 2 2 1 Ifq=3x +6x -7x +πx - √2x x +8x x +πx x + 10x x - x x 1 2 3 4 1 2 1 3 3 4 2 4 2 1 4 then the matrix for q is 3-√2/2 4 -1/4 A= -√2/2 6 0 5 4 0 -7 π/2 -1/4 5 π/2 π warning In example 1, do not writeq=A.Thequadratic form does not equal a matrix (q is x1 Û T Û Û a scalar quantity, not a matrix). What q does equal is x Ax where x = » x4 example 2 23 5 1 If q has matrix 37-2 thenq=2x2 +7y2 + z2 + 6xy + 10xz - 4yz. 2 5 -2 1/2 page 2 of Section 9.1 basis changing rule for the matrix of a quadratic form Suppose q is a quadratic form in variables x,y,z with (old) matrix A. Let u=u1i+u2j+u3k v=v1i+v2j+v3k w=w1i+w2j+w3k be a new basis for R3. Let P be the usual basis—changing matrix: u v w 1 1 1 P= u2 v2 w2 u3 v3 w3 Then the new matrix for q in the new coord system with variables X,Y,Z and basis vectors u,v,w is given by new=PT old P example 3 Suppose (3) q = x2 +6xy+y2. Let u= i+2j v=-i+2j Find the matrix for q w.r.t. basis u,v and express q in terms of new coordinates X and Y. solution method 1 (using an algebraic substitution) Let 1-1 P= 22 The old coordinates x,y and new coordinates X,Y are related by x X []y =P[]Y so x=X-Y y=2X+2Y Substitute for x and y in (3): q = (X-Y)2 + 6(X-Y)(2X+2Y) + (2X+2Y)2. Multiply out and collect terms to get q in terms of X and Y: q = 17X2 +6XY-7Y2. 17 3 The matrix for q w.r.t. basis u,v is . 3-7 method 2 (using the basis changing rule for the matrix of a quadratic form) 13 The matrix for q (w.r.t. the standard coord system) is A = []31 . By the basis changing rule above, the new matrix for q is page 3 of Section 9.1 12 13 1-1 17 3 PT AP = = -1 2 31 22 3-7 Use the matrix to express q in terms of X and Y: q = 17X2 +6XY-7Y2 proof of the basis changing rule x x T x q=[]xyz A y = y A y z z z X T X by the basis changing rule = P Y AP Y for coords of a vector Z Z X T = []XYZ PAP Y by T rule Z So PTAP satisfies (1) but using new coordinates. Furthermore PTAP is symmetric because (PTAP)T =PTATPTT =PTAP (since AT =A) So PTAP is the matrix of q w.r.t. the new basis. warning 1. The matrix of a quadratic form must be symmetric. 2. If A is symmetric then PTAP is also symmetric. If yours isn't, check your arithmetic. congruent matrices (versus similar matrices) If A is symmetric and P is invertible (so that its cols are independent and can T serve as basis vectors) then P AP and A are called congruent. The matrix PTAP represents the same quadratic form as A, but w.r.t. a new basis consisting of the cols of P. -1 Remember from Section 6.2 that P AP and A are called similar (whether or not A is symmetric). The matrix P-1AP represents the same linear transformation as A but w.r.t. a new basis consisting of the cols of P. If the matrix P is orthogonal, which happens when the new basis is orthonormal, then PT =P-1 and congruence is the same as similarity. mathematical catechism question 1 What is a quadratic form. answer A quadratic form say in variables x1,x2,x3,x4 is an expression of the form 2 2 2 2 a1x1 +a2x2 +a3x3 +a4x4 + cross terms like b1x1x2,b2x1x3 etc. question 2 What is the matrix of a quadratic form. answer The matrix of a quadratic form in variables x1,..., xn is the symmetric page 4 of Section 9.1 x1 matrix A such thatq=[x ... x ]A » 1 n xn question 3 What are congruent matrices. answer Matrices A and B are congruent if A is symmetric and there exists an invertible matrix P such thatB=PTAP (this automatically makes B symmetric too). question 4 What are similar matrices. answer Matrices A and B are similar if there exists a matrix P such that B=P-1AP. PROBLEMS FOR SECTION 9.1 1.Letq=x2 +3y2 +8z2 - 3xy - 4yz. Find the matrix A for q and write q in terms of A using matrix notation. 234 2. Write out the quadratic form which has matrix 367. 479 page 5 of Section 9.1 2 3. Supposeq=x2 +3x4 x5. Find the matrix for q. 12 3 4 25 0 0 4. Find the quadratic form with matrix . 300-3 40-36 5. Let q=x2 +4xy+3y2 Û Û Û u =3i+j Û Û Û v =2i-j. (a) Find the new formula for q w.r.t. the basis u,v using the basis changing rule for a quadratic form. (b) Find the new formula for q w.r.t. the basis u,v again using an algebraic substitution. (c) Suppose a point has coords X=1, Y=2 w.r.t. basis u,v. Find the value of q at the point two ways, using its X,Y coordinates and then again using its x,y coordinates. 6. Start withq=x2 +3xy-5y2 and make the change of variable X=x-y Y=x+y (a) Find q in terms of X and Y just with algebra. (b) What new basis is involved when you use variables X and Y. (c) Find q in terms of X and Y again using the basis changing rule for q. (d) Find the (old) matrix for q. What is the connection between q and the old matrix (write an equation beginning "q = "). (e) Find the new matrix for q. What is the connection between q and the new matrix (write an equation beginning "q = "). (f) What's the connection between the old matrix for q and the new matrix for q. 7. Start withq=x2 +4xy-y2 and make the change of variable x=2X-Y y= X+3Y (a) Find q in terms of X and Y just with algebra. (b) What new basis is involved when you use coordinates X and Y. (c) Find q in terms of X and Y again using the basis changing rule for q. 8. Suppose the X—axis is the same as the x—axis and the Y—axis is found by rotating the y—axis clockwise 45o. Use the basis changing rule for a quadratic form to find (a) the new formula forq=x2 +y2. (b) the old formula forq=XY. (c) the equation of the circle x2 +y2 = 1 in the new system. 9. In the usual x,y coord system, q is 2x2 +3xy+4y2. Switch to a new X,Y coord system which has the same axes as before but new scales. If the old scale was the inch, on the new X-axis use the foot and on the new Y-axis use a half—inch. Find q in the new coord system. page 1 of Section 9.2 SECTION 9.2 DIAGONALIZING A QUADRATIC FORM diagonalizing q Start with a quadratic form q, in say 3 variables, with matrix A. Diagonalizing q means finding a new X,Y,Z coord system in which the formula for q has no cross terms, i.e., is of the form aX2 +bY2 +cZ2. Equivalently, diagonalizing q means finding an invertible matrix P so that the PTAP, the new matrix for q, is diagonal. first method for diagonalizing q: using eigenvalues Suppose q is a quadratic form in variables x,y,z with matrix A. Since A is symmetric it has a complete set of orthonormal eigenvectors with corresponding real 3 eigenvalues ¬1,¬2,¬3. Then w.r.t. a basis for R of orthonormal eigenvectors of A, 2 2 2 q=¬1 X +¬2 Y +¬3 Z In other words, the new matrix for q w.r.t. the new basis is ¬ 00 1 Ò= 0¬2 0 00¬3 proof Let u,v,w be the orthonormal eigenvectors and let P be the matrix with cols u,v,w.