Salmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2

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1 Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to some nonlnear equatons), and seems to have been motvated by the success of the theory of frst-order PDEs. In ths method one rewrtes the hgher order PDE as a system of frst-order PDEs and attempts to generalze the method of characterstcs to that system. Ths turns out to be possble only for a restrcted (but mportant) class of PDEs called hyperbolc. The other classfcaton method apples only to lnear, second-order equatons. Ths lecture covers ths second method; we postpone the frst method untl the last lecture. Consder the general lnear, second-order PDE n the form (1) A θ θ + B x x + Cθ F( x) x where θ s a functon of the n varables x ( x 1,x,K, x n ). We assume that all the coeffcents A, B and C are constants. Wth no loss n generalty, we assume that A s symmetrc. We wll show that, because A s symmetrc, t s possble to transform to new coordnates n whch (1) takes the dagonal form () θ A x + θ B + Cθ F( x) x and each A takes the value +1, 1 or 0. For example, the general -dmensonal equaton (3) A 11 + A 1 θ xy + A θ yy +l.o.d. F (where l.o.d. means lower-order dervatves ) can always be transformed nto one of the followng forms: (4) +θ yy + l.o.d. F θ yy + l.o.d. F ±θ y + l.o.d. F plus varous permutatons of these, such as θ yy ± θ x + l.o.d. F. In (4) we wrte only the hghest-dervatve terms n each varable, and we do not consder the case θ x +θ y +L because t s a frst-order equaton. It turns out that some of the most mportant propertes of such equatons (such as the approprate form of boundary condtons) depend only on the hghest-dervatve terms. Thus there s some pont n consderng the cases (4) n ther purest forms : +θ yy F Posson s equaton 5-1

2 Salmon: Lectures on partal dfferental equatons (5) θ tt F wave equaton θ t F heat equaton (where the notaton hnts at the typcal physcal meanng). To prove our clam we must frst prove a theorem about symmetrc A. For future purposes, we shall prove more than s actually needed for ths lecture. Theorem. [The spectral decomposton theorem.] If the matrx A s Hermtan (meanng that A A * where * denotes the complex conugate), then all the egenvalues of A are real, and all the egenvectors are, or can be made, orthogonal. Note that real symmetrc A, lke that n (1), are a class of Hermtan matrces. Before provng ths theorem, we note that the problem of dagonalzng (1) s closely related to the problem of dagonalzng the quadratc form (6) A x + B x + Cθ. In both problems the strategy s the same: choose the new coordnates x to be the coeffcents n the expanson (7) x x e of x n the n egenvectors e of A. Then the frst term n (6) becomes (8) x T Ax e x x e T Ae, T A x e ( ) x ( ) x λ e e e T λ e, λ x, where λ s the egenvalue correspondng to egenvector e. Note that we use the supposed orthogonalty of the egenvectors n ways: frst, n the assumpton (7) that any x can be expanded n these egenvectors, and, second, n the fnal step of (8). By an addtonal rescalng of x, we may reduce (8) to smplest form, x. The transformaton of (1) to the dagonal form () proceeds n the same manner. But frst we prove the theorem. Sketch of the proof. Recall that the column vector e s an egenvector of the n n matrx A f Aeλe where λ s the correspondng egenvalue, a generally complex number. Thus the egenvalues of A correspond to the roots of det (A λi) 0. Let λ be the egenvalue of A correspondng to egenvector e : (9) Ae λ e. Let A + A T * (transpose conugate). Thus A A + f A s Hermtan. 5-

3 Salmon: Lectures on partal dfferental equatons To prove that λ s real, take the transpose conugate of (9) to get (10) e + A + λ * e +. (Note that e + s a row vector.) Now multply (9) on the left by e +, multply (10) on the rght by e, and subtract, usng the fact that A A +. The result s * (11) ( λ λ )e + e 0 whch proves that λ s real. To prove that the e are orthogonal, consder any egenvalues λ and λ. Multply (9) by e +, multply (1) e + A + λ e + (obtaned by takng the Hermtan conugate of (9)) by e on the rght and subtract, agan usng the fact that A s Hermtan, to obtan (13) ( λ λ )e + e 0. Thus f λ λ, then (14) e + e e e 0 and the two egenvectors are orthogonal. Ths proof does not apply to the case λ λ of degenerate egenvalues, whch we postpone untl later n ths lecture. Assume for the moment that the n egenvalues are dstnct. Then we have n orthogonal egenvectors. These can easly be normalzed; we henceforth assume that they are. The normalzed egenvectors can be used to dagonalze A. Consder the n n matrx (15) U ( e 1,e,K,e n ) whose column vectors are the orthonormal egenvectors of A. Then (16) U + U 1, that s, (17) U + U I whch smply restates the fact that the e are orthonormal. Matrces U wth the property (17) are called untary. [Note that snce A and λ are real, t s always possble to choose 5-3

4 Salmon: Lectures on partal dfferental equatons real egenvectors e. Then U s real and U + U T ]. The matrx U defned by (15) dagonalzes A n the sense that (18) U 1 AU U T AU U T dag( λ 1,λ,K, λ n ) ( ) λ 1 e 1,λ e,k,λ n e n Now we return to the general equaton (1) wth second-order term (19) A θ x Defnng U as above, we ntroduce the new varables (0) x U T. Thus x s the proecton of x onto e ; (0) s equvalent to (7). Snce (1) x x T U we have () θ A A x x U T T k U m km θ θ A x k x km m km x k x m where, n matrx notaton, (3) A U T AU. But, by (18), A s the dagonal matrx wth the egenvalues of A on the dagonal. Thus (4) A θ x θ λ x. Ths s almost the smplest form. To obtan the smplest form, t s only necessary to redefne (5) x x / λ (provded that λ 0.) In overall summary, to transform the second dervatves n (1) to the canoncal form (), we use the transformaton (6) x U T x 5-4

5 Salmon: Lectures on partal dfferental equatons where U s the untary matrx defned by (15). The components of x are ust the ampltude of each egenvector s contrbuton to x. Snce untary transformatons have the property that they preserve x x (prove ths!), the transformaton (6) may be nterpretted as a rotaton of the coordnates n n-dmensonal space. How does ths work n the case n? We let a A 11, b A 1, and c A, so (1) takes the form (7) a + bθ xy + cθ yy +L F The egenvalues of (8) A a b b c satsfy (9) ( a λ) ( c λ) b. Thus (30) λ a + c ± 1 ( a c) + 4b a + c ± 1 from whch we see that λ s ndeed real for any a, b, c. If (31) b ac 0 (the parabolc case) ( a + c) + 4( b ac) t follows from (30) that at least one of the egenvalues vanshes. Ths s the parabolc case and leads to +θ y +L or θ yy + θ x +L Smlarly, the egenvalues have the same sgn f (3) b ac < 0 (the ellptc case) Ths s the ellptc case and leads to +θ yy +L Fnally, the egenvalues have opposte sgns f (33) b ac > 0 (the hyperbolc case) Ths s the hyperbolc case and leads to θ yy +L We have already mentoned the connecton wth quadratc forms. For the form ax + bxy + cy, the same transformaton produces a dagonalzaton, and the resultng equaton descrbes a parabola, ellpse or hyperbola. Quadratc forms le at the heart of Remannan geometry. There one seeks coordnates n whch the dfferental arc length ds, whch has the general form 5-5

6 Salmon: Lectures on partal dfferental equatons (34) ds smplfes to A dx d (35) ds ( dx 1 ) ± ( dx ) L ± ( dx n ) The new coordnates are called Cartesan coordnates. Examples nclude Eucldean space ds ( dx) + ( dy) + ( dz) Lorentzan spacetme ds c ( dt) ( dx) ( dy) ( dz) If you have studed general relatvty (where A s always wrtten as g ) then you know that global Cartesan coordnates exst only f the curvature tensor assocated wth g vanshes. Ths bears upon the generalzaton of (1) to the case of nonconstant coeffcents A ( x). Whereas (1) can always be transformed locally to the form (), there s n general no globally vald transformaton. For the case n however, one can show that a global transformaton to the form () s possble provded that the A do not vary n such a way that the equaton changes type (e.g. from ellptc to hyperbolc). For a careful dscusson of ths, see Garabedan, Chapter. We must stll complete the proof of the spectral decomposton theorem by showng that orthonormal egenvectors may be found even n the case where some of the egenvalues are equal. Completon of the proof. Even f all the egenvalues are equal, we can fnd at least one egenvector e 1. Then we defne ( ) (36) U 1 e 1, e ˆ,K, ˆ e n where e ˆ,K, ˆ e n are n 1 other vectors (not necessarly egenvectors of A) whch are orthonormal to e 1 and to each other. (It s always possble to fnd such a set!) By the orthonormalty of ts columns, U 1 s untary. We use U 1 to transform A nto the form λ 1 0 L 0 0 (37) U 1 1 AU 1 M A 0 where A s an ( n 1) ( n 1) matrx. Then we do the same thng to A. That s, we fnd the (untary) U such that 5-6

7 Salmon: Lectures on partal dfferental equatons λ 1 0 L 0 0 λ 0 L 0 (38) U 1 ( U 1 1 AU 1 )U 0 M M A and keep gong. Each step corresponds to a rotaton n the remanng coordnates. At the end we have (39) U 1 AU dag( λ 1,λ,K, λ n ) where U U 1 U LU n s untary because the product of untary matrces s untary (prove ths!). Eqn (39) mples that the columns of U are the sought-for orthonormal egenvectors. (Each U has rank n, hence so does U.) References. Mathews and Walker Chap 6, Zauderer Chap