Diagonalization of Quadratic Forms. Recall in days past when you were given an equation which looked like

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1 Diagoalizatio of Qadratic Forms Recall i das past whe o were gie a eqatio which looked like ad o were asked to sketch the set of poits which satisf this eqatio. It was ecessar to complete the sqare so that the eqatio looked like the (h,k) form of a ellipse. That is, ( ) ) ( ) ( ) 4 The, ow that we hae rewritte the eqatio ito a form we recogize, we ca see that this is a circle (which is a ellipse) of radis 4,. Bt we had to do some work to get to this poit. Similarl, we eed to be able to rewrite a b c ito a ew form so that we ca read off all of the iformatio ecessar to sketch the set. cetered at the poit ( ) Sppose that we are gie a eqatio of the followig form, a b c. The, as it trs ot, we ca rewrite this eqatio sig matri otatio. Ideed, a b a b c [ ]. b c a b The ice thig abot this represetatio is that the matri has a propert called b c smmetr. Specificall the matri is smmetric abot the diagoal. Formerl, T a b a b. There are ma ice reslts abot smmetric matrices. Oe of them b c b c is the fact that a smmetric matri is itaril diagoalizable. Oka, so I m throwig ot a lot of fac words, so let me tr to elaborate o eactl what this meas. Defiitio : (Uitar atri) A matri U is said to be itar if the colms of U form a orthoormal basis for R. Defiitio : (Uitaril Diagoalizable) A matri A is said to be itaril diagoalizable if there eists a itar matri U so that A UΛU where Λ is a diagoal matri. Recall : A set of ectors { e, e,, } ector e e i j R K e is a orthoormal basis for α e α e α e ad has a (iqe) represetatio i j. i j R if ad ol if a

2 The with these facts it shold become clear as md that T U U. Care to see the proof? Too bad, I m doig it awa. Propositio : If U is a itar matri the, T I U U, where I deotes the idetit matri. Ths, U U T. Proof : et U be a itar matri. The obsere the followig comptatio. T U U O The sice the i s form a orthoormal basis we hae, Ι O O as desired. So, to traslate this ito slightl more derstadable lagage, this meas that a itar matri is a matri which moes arod the coordiate aes to a ew set of orthogoal aes (i.e. a chage of basis). Or, we cold sa that the aes hae bee rotated (or reflected) to a ew set of perpediclar aes. Allow me to go back to a preios eample to tr to lower the leel of itimidatio a little bit. Recall the ol complete the sqare trick to fid the (h,k) form for a eqatio represetig a ellipse. This represetatio correspods to shiftig the origi to the poit (h,k) ad drawig the ellipse arod that poit, right? So, if o feel reasoabl comfortable shiftig arod the coordiate aes, which is eactl what o are doig whe completig the sqare, the o shold feel comfortable twirlig the aes arod. After all it s the same idea. The ol differece is that this ew tool we hae deeloped is more comptatio itesie. et s ow set p a set of procedres to perform this task. ) Write c b a i the matri form [ ] c b b a. ) Fid the eigeales ad for the matri c b b a. ) Fid the eigeectors ad correspodig to the eigeales ad respectiel. 4) Draw lies passig throgh the ectors ad. (These are the ew aes.) ) Rewrite the ew eqatio as w w where w ad w are ariables represetig the distace from the origi i the ad directios

3 respectiel. Jst like ad represet the distace from the origi i the directio e ad e. 6) Draw the dar pictre. I ca hear o ow, Whoa hoss! Yo did t do that itar diagoalizatio thig o were talkig abot! Yo re right. To sketch the pictre I did t eed to go that far. Howeer, otice that I hae all the iformatio to do so if I wish. et ad. The, a b is the desired itar diagoalizatio. b c

4 Eample : Sketch the image of the set of poits which satisf the eqatio 8 7. Soltio : ) [ ] ) 7 det ( )( ) ( ), ) 7 7 is a eigeector for. 7 7 is a eigeector for.

5 4) Sketch o a piece of paper lies passig throgh ad. These represet the ew aes. Notice. w ) The ew eqatio we hae is the w. Notice this is a bit of a weird aimal. What does it sa? Well, doig some algebra we hae, w w w w ± This gies s a set of parallel lies which itersect the w ais at ±. 6) Draw the dar pictre.

6 Eample : Sketch the image of the set of poits which satisf the eqatio. Soltio : ) [ ] ) det ( )( ) ( )( ) 8 8, ) 8 8 is a eigeector for is a eigeector for 8.

7 4) Sketch o a piece of paper lies passig throgh ad. These represet the ew aes. Notice. ) The ew eqatio we hae is the 8w, which after some cleaig p looks like w 8w w ( ) ( ). This the is a ellipse that itersects the w ais at ± ad itersects the w ais at ± 6) Draw the dar pictre.

8 Eample : Sketch the image of the set of poits which satisf the eqatio 4. Soltio : 4 ) 4 [ ] 4 ) det 4 ( )( ) 4, Ick. How ca we possible fid eigeales correspodig to these mosters? Well, o ca if o wat bt o do t eed to. Becase if we skip the whole fid the eigeectors step otice what will happe. That is, assme for the momet that we fod the eigeectors for these eigeales. The we hae the ew represetatio for the eqatio as follows, w w w w Do o see athig wrog with this pictre. Notice, >, >, w >, ad w > bt the eqatio sas that I ca mltipl them together ad add them p to obtai a egatie mber. There are certail o real mbers which ca satisf this eqatio. So, the sketch of the set of poits which satisf the eqatio w w is empt. To smmarize the cases we hae the followig, i) If ad we hae a ellipse. > > ii) If < (so either < or <, ot both) we hae a hperbola. iii) If or (ot both zero) we hae a set of parallel lies. i) If we hae the empt set. ) If < ad we hae the empt set. <