線性代數(shù)II-chapter5(1-2).PPT

1、Chapter 5 A quadratic form5.3 A positive definite quadratic form 5.1-5.2 A quadratic form and its standard formA quadratic form A standard form Congruence 122221211 1222121213 131123231,A homogeneous second degree polynomial in unknowns , 222 22nnnnnnnnnnnx xxf x xxa xa xa xa x xa x xa x xa x xax1is

2、 called a quadratic form.nx njijiijnnnnnnnnnnnnjiijxxaxaxxaxxaxxaxaxxaxxaxxaxaxxxfnjiaa1,222112222221221112112211121 ,then, 2 , 1, If 1 nnnnnnnnnnnnnnnnnnnnnxaxaxaxaxaxaxaxaxaxxxxaxaxaxxaxaxaxxaxaxaxxxxf221122221211212111212211222212121212111121, ,is that form,matrix a into form quadratic thecan wri

3、te We2111211212222121211112122122212 , where ,and is called the matrix of the quadrnnnnnnnnnnTnnnnnaaaxaaaxx xxaaaxxaaaxaaax AxxAxaaaAatic form,where is a symmetric matrix.TAA222123123121323Express the following quadratic form into a matrix form,324f x x xxxxx xx xx x21231121322122323 1323Since1,32

4、+21 22fx xxxx xx xx xxx xx xx xx112312323thus, 1312,1121212xf x x xx x xxxFor given matrices and , if there is an invertiblematrix , such that then is ongruence to , denoted by .Tn nABPP APBAcBAB 1 Obviously, the following properties are satisfied(I) ; (II) If , then ;(III) If , then .AAABBAAB BCAC

5、2 The quadratic form can be changed into, under the linear transformation there is an invertible matrix such that ,that is TTTfx Axfy ByxCyCBC ACAB2221122The quadratic form is called the standard form.nnfd yd yd y ifonly and if n nsformatiolinear tra under the form satndard theinto form quadratic th

6、echange To 12222211CyxydydydfAxxfnnT,diagsuch that matrix invertiblean find is,that , 212121212222211nTnnnnnTTdddACCCyyydddyyyydydydyACCyfThere are three methods to transform a quadratic form into a standard form., of eignvalues are , where such that matrix) orthogonalan is ( ation transformorthogon

7、alan is there, form quadraticany For 2122222111,AyyyyAPPyAxxfPPyxAxxxxafnnnTTTTnjijiijly.respective , to ingcorrespond of seignvector are ,then , let if212121nnnAppppppPThe proof of the above theorem is straightforward and is provided here for the interested students.(1) Rewrite the quadratic form i

8、nto the matrix form; 122 Find eigenvalues , of ;nA AAAxxxxafTTnjijiij where, is, that 1, 123 Find corresponding eigenvectors ,of every eigenvalue ;rii 124 Orthogonalize eigenvectors , using the Gram-Schmidt Method;ri 121211222211225 Obtain a orthonormal eigenvectors ,let , then diag ,that is, under

9、the orthognnTnnnPP APP APfyyy onal transformation .xPy22212323Transform a quadratic form 4332into the standard form.fxxxx x ,310130004 is ofmatrix TheAf242310130002then AE123So the eigenvalues of are 2,4.A , 02 equations of systemlinear shomogeneou thesolve we, 2 eigenvalue toingcorrespondr eigenvec

10、toan find To1xAE123that is, 200 0110, 011we reduce the coefficient matrix to row echelon formxxx111111to get a solution 0,1, 1, which is all eigenvectors corresponding to eigenvalue 2,then we take an unit eigenvector, 22 0,22TTkp23123For 4, we solve the homogeneous linear system of equations 40, 000

11、that is 0110,011EA xxxx2323we obtain a set of vectors 1,0,0,0,1,1is a maximal set of linearly independent eigenvectorscorresponding to eigenvalue 4,TT222333 then we take nuit eignvectors 1,0,0,22 0, 22thus, under the orthogonal transformation TTpp112233222123010220 , 2222022we have 244xyxyxyfyyyis?1

12、662355surface quadratic thesurfaceWhich 323121232221xxxxxxxxxf ,333351315 is ofmatrix TheAf94333351315then AE123So the eigenvalues of are 0,4,9.By theorem 1,A2223 can be transformed into the standard form 49 under the orthogonal tranformatiom.ffyy222123121323Since the orthogonal transformation does

13、not changethe shape of a surface in space , so 5532661 is an elliptic cylinder.fxxxx xx xx xnTAPPAPPPA,diag such that matrix orthogonalan exists there,matrix symmetric realgiven any for is, that able,diagonaliz bemust matrix symmetric realany know, all weAs11So we have the following resultsuch that

14、, matrices elementary of sequence a is there,matrix symmetric realany For 21sPPPA, diag212111nsTTsTsPPAPPPPIn summary, the following is a procedure for diagonalizing a symmetric matrix, that is, the procedure of elementary transformation., diag where , thus, diagSince212121212111211121nPPAPPPPPPEPss

15、nsTTsTsdddCEAPPEPPPPCdddPPAPPPPsTTsTssCyxxxxxxxxxf transformliear invertible thefind and form, standard theinto 4642 Thransform2332223121 ,432311210 is ofmatrix TheAf1003110101350510001100001010423201311100010001432311210 13121312212133rrrrccccrrccEACyxyyyfCCrrccunder 12,100211510,1200010001 where,1002115101200010001 232221552323Sometimes, we also use the elementary operations to collocate the quadratic form into the standard form. The following example uses this method.Cyxxxxxxxxxxf transformliear invertible thefind and form, standard theinto 62252 Thransform323