Question Number 212028 by siva12345 last updated on 27/Sep/24
![DEFINATION OF QUADRATIC FORM: A Quadratic form is a homogeneous polynomial of degree two in multiple variable. Q=X^T AX Here Q=Quadratic form. ax^2 +by^2 +cz^2 +2hxy+2fyz+2gzx=0 By using these Q=X^T AX [we can write matrix A] A= [(x),(y),(z) ] [((a h g)),((h b f)),((g f c)) ] [(x,y,z) ] By using the characteristic equation of matrix ∣A−λI∣ =0 To get eigen equation to can solve eigen equations to get eigen values. Suppose substitute the eigwn value in ∣A−1λI∣X =0 After row transformation we can get x_1 ,x_2 ,x_3 P =[e_(1 ) e_2 e_3 ] ∴ {Here p is a modal matrix} Since p is a orthogonal matrix p^(−1) =p D=P^T AP Where D is diagonal matrix. The quadratic form can be reduce into normal form X = Y^T DY. Here: Index = The number of positive terms in its canonical form Signature = The difference of positive and negative term in the canonical form If all λ>0⇒ positive definite. If all λ<0⇒ negative definite. If all λ≥0 atleast one λ= 0⇒positive semidefinite. If all λ≤ 0 atlest one λ= 0⇒negative semidefinite. If some λ are positive and some λ are negative⇒indefinite. X = PY is used transformation of quadratic form to normal form (or) canonical form. Eg: (Q) Reduce the quadratic form 3x^2 +2y^2 +3z^2 −2xy−2yz = 0 to canonical form by orthogonal transformation and also find rank, index, nature and signature. Sol: The given equation is 3x^2 +2y^2 +3z^2 −2xy−2yz=0 3xx+2yy+3zz−xy−xy−yz−yz=0 we know that Q=X^T AX = [(X),(Y),(Z) ] [(( 3 −1 0)),((−1 2 −1)),(( 0 −1 3)) ] [(X,Y,Z) ] The corresponding symmetric matrix A= [(( 3 −1 0)),((−1 2 −1)),(( 0 −1 3)) ] To find the eigen values we can use the chareteristic eauation of matrix A is ∣A−λI∣= 0 [(( 3−λ −1 0)),((−1 2−λ −1)),(( 0 −1 3−λ)) ]= 0 (3−λ)[(2−λ)(3−λ)−1)]+1[−1(3−λ)]= 0 (3−λ)(−5λ+λ^2 +5)+(λ−3)= 0 λ^3 −8λ^2 +19λ−12= 0 λ=1,3,4 Case i: λ=1 substitute in [A−λI]X = 0 [(( 2 −1 0)),((−1 1 −1)),(( 0 −1 2)) ] [(x_1 ),(x_2 ),(x_3 ) ]= [(0),(0),(0) ] R_2 ⇒R_2 +2R_1 [(( 2 −1 0 )),(( 0 1 −2)),(( 0 −1 2)) ] [(x_1 ),(x_2 ),(x_3 ) ] = [(0),(0),(0) ] R_3 ⇒R_3 +R_2 [((2 −1 0 )),((0 1 −2)),((0 0 0)) ] [(x_1 ),(x_2 ),(x_3 ) ] = [(0),(0),(0) ] 2 x_(1 ) −x_2 =0 x_2 −2x_3 =0 x_3 = k x_2 = 2k x_1 = k x_1 = [(1),(2),(1) ]k Case ii: λ = 3 To substitite λ value in [A−λI]X = 0 [(( 0 −1 0)),((−1 −1 −1)),(( 0 −1 0)) ] [(x_1 ),(x_2 ),(x_3 ) ]= [(0),(0),(0) ] (R_1 ⇔R_2 ) [(( −1 −1 −1)),(( 0 −1 0)),(( 0 −1 0)) ] [(x_1 ),(x_2 ),(x_3 ) ]= [((0 )),(0),(0) ] R_3 ⇒R_3 −R_2 [(( −1 −1 −1)),(( 0 −1 0)),(( 0 0 0)) ] [(x_1 ),(x_2 ),((x3)) ] = [(0),(0),(0) ] x_2 = 0 −x_1 −x_2 −x_(3 ) =0 x_3 = k x_(1 ) = −k x_2 = [((−1)),(( 0)),(( 1)) ]k Case iii: λ = 4 To substitute in equation [ A−λI ]X = 0 [(( −1 −1 0)),(( −1 −2 −1)),(( 0 −1 −1)) ] [(x_1 ),(x_2 ),(x_3 ) ] = [(0),(0),(0) ] R_2 ⇒R_2 −R_1 [(( −1 −1 0)),(( 0 −1 −1)),(( 0 −1 −1)) ] [(x_1 ),(x_2 ),(x_3 ) ] = [(0),(0),(0) ] R_3 ⇒R_3 −R_2 [(( −1 −1 0)),(( 0 −1 −1)),(( 0 0 0)) ] [(x_1 ),(x_2 ),(x_3 ) ] = [(0),(0),(0) ] −x_1 −x_(2 ) = 0 −x_3 −x_3 = 0 x_(3 ) = k x_2 = −k x_1 = k x_(3 ) = [(( 1)),((−1)),(( 1 )) ]k we can observe that vectors are orthogonally mutually we normalize this vectors and obtain. e_1 = [((1/( (√6)))),((2/( (√6)))),((1/( (√6)))) ] e_2 = [(((−1)/( (√2)))),(( 0)),(( (1/( (√2))))) ] e_3 = [((1/( (√3)))),(((−1)/( (√3)))),((1/( (√3)))) ] let P = the modal matrix in normalized form P = [e_(1 ) e_2 e_3 ] = [(((1/( (√6))) ((−1)/( (√(2 )))) (1/( (√3))))),(((2/( (√(6 )) )) 0 ((−1)/( (√3))))),(((1/( (√6))) (1/( (√2))) ((−1)/( (√3))))) ] Since p is a orthogonal matrix P^(−1 ) = P^T So D = P^T AP where D is diagnol matrix = [(((1/( (√6))) ((−2)/( (√6))) (1/( (√6))))),((((−1)/( (√2))) 0 (1/( (√2))))),(((1/( (√3))) ((−1)/( (√3))) ((−1)/( (√3))))) ] [(( 3 −1 0)),((−1 2 −1)),(( 0 −1 3)) ] [(((1/( (√(6 )))) −(1/( (√2))) (1/( (√3))))),(((2/( (√6))) 0 ((−1)/( (√3))))),(((1/( (√6))) (1/( (√2))) −(1/( (√3))))) ] D = [((1 0 0 )),((0 3 0)),((0 0 4)) ] The quadratic form can be reduce to normal form Y^T DY Y^( T) DY = [y_1 y_(2 ) y_3 ] [((1 0 0)),((0 3 0)),((0 0 4)) ] [(y_1 ),(y_2 ),(y_3 ) ] y_1 ^2 +3y_2 ^2 +4y_3 ^(2 ) = 0 Rank = 3 Index = 3 Signature= 3 Nature=Positive definite By orthogonal tfansformation X = PY [(x_1 ),(x_2 ),(x_3 ) ] = [(((1/( (√6))) ((−1)/( (√2))) (1/( (√3))))),(((2/( (√6))) 0 ((−1)/( (√3))))),(((1/( (√(6 )))) (1/( (√2))) (1/( (√3))))) ] [(y_1 ),(y_2 ),(y_3 ) ] x_(1 ) = (y_1 /( (√6)))−(y_2 /( (√2)))+(y_3 /( (√3))). x_2 = ((2y_1 )/( (√6)))−(y_3 /( (√(3.)))) x_3 = (y_1 /( (√6)))+(y_2 /( (√2)))+(y_3 /( (√(3.)))) o](https://www.tinkutara.com/question/Q212028.png)
$${DEFINATION}\:\:\:\:{OF}\:\:\:{QUADRATIC}\:\:{FORM}:\: \\ $$$$\:\:\:\:\:{A}\:\:{Quadratic}\:\:{form}\:\:{is}\:\:{a}\:{homogeneous}\:\:{polynomial}\:\:{of}\:\:{degree}\:{two}\:\:{in}\:\:{multiple}\:\:{variable}.\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{Q}={X}^{{T}} {AX} \\ $$$${Here}\:\:{Q}={Quadratic}\:{form}. \\ $$$${ax}^{\mathrm{2}} +{by}^{\mathrm{2}} +{cz}^{\mathrm{2}} +\mathrm{2}{hxy}+\mathrm{2}{fyz}+\mathrm{2}{gzx}=\mathrm{0} \\ $$$${By}\:\:{using}\:\:{these}\:\:{Q}={X}^{{T}} {AX}\:\:\left[{we}\:\:{can}\:\:{write}\:{matrix}\:{A}\right] \\ $$$${A}=\begin{bmatrix}{{x}}\\{{y}}\\{{z}}\end{bmatrix}\begin{bmatrix}{{a}\:\:\:\:\:{h}\:\:\:\:\:{g}}\\{{h}\:\:\:\:\:{b}\:\:\:\:\:{f}}\\{{g}\:\:\:\:\:{f}\:\:\:\:\:{c}}\end{bmatrix}\begin{bmatrix}{{x}}&{{y}}&{{z}}\end{bmatrix} \\ $$$${By}\:\:{using}\:\:{the}\:\:{characteristic}\:\:{equation}\:\:{of}\:\:\:{matrix}\:\:\mid{A}−\lambda{I}\mid\:=\mathrm{0} \\ $$$${To}\:\:\:{get}\:\:{eigen}\:\:{equation}\:\:\:{to}\:\:{can}\:\:{solve}\:\:{eigen}\:\:{equations}\:\:{to}\:\:{get}\:\:{eigen}\:{values}. \\ $$$${Suppose}\:\:{substitute}\:\:{the}\:\:{eigwn}\:\:{value}\:\:{in}\:\:\mid{A}−\mathrm{1}\lambda{I}\mid{X}\:=\mathrm{0} \\ $$$${After}\:\:{row}\:\:{transformation}\:\:{we}\:\:{can}\:\:{get}\:\:{x}_{\mathrm{1}} ,{x}_{\mathrm{2}} ,{x}_{\mathrm{3}} \\ $$$${P}\:=\left[{e}_{\mathrm{1}\:\:} {e}_{\mathrm{2}} \:\:{e}_{\mathrm{3}} \right]\:\:\:\:\:\:\:\:\therefore\:\left\{{Here}\:\:{p}\:\:{is}\:\:\:{a}\:\:{modal}\:\:{matrix}\right\} \\ $$$${Since}\:\:{p}\:\:{is}\:\:{a}\:\:{orthogonal}\:\:{matrix}\:\:{p}^{−\mathrm{1}} ={p} \\ $$$${D}={P}^{{T}} {AP}\:\:\:\:\:{Where}\:\:{D}\:\:\:{is}\:\:{diagonal}\:\:{matrix}. \\ $$$${The}\:\:{quadratic}\:\:{form}\:\:{can}\:\:{be}\:\:{reduce}\:\:{into}\:\:{normal}\:\:{form}\:\:{X}\:=\:{Y}^{{T}} {DY}. \\ $$$${Here}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{Index}\:\:\:\:=\:\:{The}\:{number}\:{of}\:{positive}\:{terms}\:{in}\:{its}\:{canonical}\:\:{form} \\ $$$${Signature}\:=\:\:{The}\:{difference}\:{of}\:{positive}\:{and}\:{negative}\:{term}\:{in}\:{the}\:{canonical}\:{form} \\ $$$${If}\:{all}\:\lambda>\mathrm{0}\Rightarrow\:{positive}\:\:{definite}. \\ $$$${If}\:{all}\:\lambda<\mathrm{0}\Rightarrow\:{negative}\:\:{definite}. \\ $$$${If}\:{all}\:\lambda\geq\mathrm{0}\:\:{atleast}\:{one}\:\lambda=\:\mathrm{0}\Rightarrow{positive}\:{semidefinite}. \\ $$$${If}\:{all}\:\lambda\leq\:\mathrm{0}\:{atlest}\:{one}\:\:\lambda=\:\mathrm{0}\Rightarrow{negative}\:{semidefinite}. \\ $$$${If}\:{some}\:\lambda\:{are}\:{positive}\:{and}\:{some}\:\lambda\:{are}\:{negative}\Rightarrow{indefinite}. \\ $$$${X}\:=\:{PY}\:{is}\:{used}\:{transformation}\:{of}\:{quadratic}\:{form}\:{to}\:\:{normal}\:{form}\:\left({or}\right)\:{canonical}\:{form}. \\ $$$${Eg}: \\ $$$$\left({Q}\right)\:{Reduce}\:\:{the}\:{quadratic}\:\:{form}\:\:\mathrm{3}{x}^{\mathrm{2}} +\mathrm{2}{y}^{\mathrm{2}} +\mathrm{3}{z}^{\mathrm{2}} −\mathrm{2}{xy}−\mathrm{2}{yz}\:=\:\mathrm{0}\:\:{to}\:{canonical}\:{form}\:{by}\:{orthogonal}\:{transformation}\: \\ $$$${and}\:{also}\:{find}\:{rank},\:{index},\:{nature}\:\:{and}\:\:{signature}. \\ $$$${Sol}:\:{The}\:{given}\:{equation}\:{is}\:\:\mathrm{3}{x}^{\mathrm{2}} +\mathrm{2}{y}^{\mathrm{2}} +\mathrm{3}{z}^{\mathrm{2}} −\mathrm{2}{xy}−\mathrm{2}{yz}=\mathrm{0} \\ $$$$\mathrm{3}{xx}+\mathrm{2}{yy}+\mathrm{3}{zz}−{xy}−{xy}−{yz}−{yz}=\mathrm{0} \\ $$$$\:{we}\:{know}\:{that}\:{Q}={X}^{{T}} {AX} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\begin{bmatrix}{{X}}\\{{Y}}\\{{Z}}\end{bmatrix}\begin{bmatrix}{\:\:\:\:\:\mathrm{3}\:\:\:\:\:−\mathrm{1}\:\:\:\:\:\:\:\:\:\:\mathrm{0}}\\{−\mathrm{1}\:\:\:\:\:\:\:\:\:\:\mathrm{2}\:\:\:\:\:\:−\mathrm{1}}\\{\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:−\mathrm{1}\:\:\:\:\:\:\:\:\:\mathrm{3}}\end{bmatrix}\begin{bmatrix}{{X}}&{{Y}}&{{Z}}\end{bmatrix} \\ $$$$ \\ $$$$\:\:\:\:\:{The}\:{corresponding}\:\:{symmetric}\:\:{matrix}\:{A}=\begin{bmatrix}{\:\:\:\:\:\mathrm{3}\:\:\:\:\:−\mathrm{1}\:\:\:\:\:\:\:\:\mathrm{0}}\\{−\mathrm{1}\:\:\:\:\:\:\:\:\:\mathrm{2}\:\:\:\:−\mathrm{1}}\\{\:\:\:\:\:\mathrm{0}\:\:\:\:\:−\mathrm{1}\:\:\:\:\:\:\:\:\mathrm{3}}\end{bmatrix} \\ $$$$\:\:\:\:{To}\:{find}\:{the}\:{eigen}\:{values}\:{we}\:{can}\:{use}\:{the}\:{chareteristic}\:{eauation}\:{of}\:{matrix}\:{A}\:{is}\:\mid{A}−\lambda{I}\mid=\:\mathrm{0} \\ $$$$\begin{bmatrix}{\:\:\:\:\:\mathrm{3}−\lambda\:\:\:\:\:−\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{0}}\\{−\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}−\lambda\:\:\:\:\:\:\:−\mathrm{1}}\\{\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:−\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{3}−\lambda}\end{bmatrix}=\:\mathrm{0} \\ $$$$\left.\left(\mathrm{3}−\lambda\right)\left[\left(\mathrm{2}−\lambda\right)\left(\mathrm{3}−\lambda\right)−\mathrm{1}\right)\right]+\mathrm{1}\left[−\mathrm{1}\left(\mathrm{3}−\lambda\right)\right]=\:\mathrm{0} \\ $$$$\left(\mathrm{3}−\lambda\right)\left(−\mathrm{5}\lambda+\lambda^{\mathrm{2}} +\mathrm{5}\right)+\left(\lambda−\mathrm{3}\right)=\:\mathrm{0} \\ $$$$\lambda^{\mathrm{3}} −\mathrm{8}\lambda^{\mathrm{2}} +\mathrm{19}\lambda−\mathrm{12}=\:\mathrm{0} \\ $$$$\lambda=\mathrm{1},\mathrm{3},\mathrm{4} \\ $$$$ \\ $$$${Case}\:{i}:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\lambda=\mathrm{1}\:{substitute}\:\:{in}\:\left[{A}−\lambda{I}\right]{X}\:=\:\mathrm{0} \\ $$$$\begin{bmatrix}{\:\:\:\:\mathrm{2}\:\:\:\:−\mathrm{1}\:\:\:\:\:\:\:\:\:\mathrm{0}}\\{−\mathrm{1}\:\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:−\mathrm{1}}\\{\:\:\:\:\:\mathrm{0}\:\:\:−\mathrm{1}\:\:\:\:\:\:\:\:\:\mathrm{2}}\end{bmatrix}\begin{bmatrix}{{x}_{\mathrm{1}} }\\{{x}_{\mathrm{2}} }\\{{x}_{\mathrm{3}} }\end{bmatrix}=\begin{bmatrix}{\mathrm{0}}\\{\mathrm{0}}\\{\mathrm{0}}\end{bmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{R}_{\mathrm{2}} \Rightarrow{R}_{\mathrm{2}} +\mathrm{2}{R}_{\mathrm{1}} \: \\ $$$$ \\ $$$$\begin{bmatrix}{\:\mathrm{2}\:\:\:\:\:−\mathrm{1}\:\:\:\:\:\:\:\:\:\mathrm{0}\:\:}\\{\:\mathrm{0}\:\:\:\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:−\mathrm{2}}\\{\:\mathrm{0}\:\:\:\:\:−\mathrm{1}\:\:\:\:\:\:\:\:\:\mathrm{2}}\end{bmatrix}\:\begin{bmatrix}{{x}_{\mathrm{1}} }\\{{x}_{\mathrm{2}} }\\{{x}_{\mathrm{3}} }\end{bmatrix}\:=\:\begin{bmatrix}{\mathrm{0}}\\{\mathrm{0}}\\{\mathrm{0}}\end{bmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{R}_{\mathrm{3}} \Rightarrow{R}_{\mathrm{3}} +{R}_{\mathrm{2}} \\ $$$$\begin{bmatrix}{\mathrm{2}\:\:\:\:\:\:−\mathrm{1}\:\:\:\:\:\:\:\:\:\mathrm{0}\:}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:−\mathrm{2}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:\:\:\mathrm{0}}\end{bmatrix}\:\begin{bmatrix}{{x}_{\mathrm{1}} }\\{{x}_{\mathrm{2}} }\\{{x}_{\mathrm{3}} }\end{bmatrix}\:=\:\:\begin{bmatrix}{\mathrm{0}}\\{\mathrm{0}}\\{\mathrm{0}}\end{bmatrix} \\ $$$$\mathrm{2}\:{x}_{\mathrm{1}\:} −{x}_{\mathrm{2}} =\mathrm{0} \\ $$$$\:{x}_{\mathrm{2}} −\mathrm{2}{x}_{\mathrm{3}} =\mathrm{0} \\ $$$$\:{x}_{\mathrm{3}} =\:{k} \\ $$$$\:{x}_{\mathrm{2}} =\:\mathrm{2}{k} \\ $$$$\:{x}_{\mathrm{1}} =\:{k} \\ $$$$\:{x}_{\mathrm{1}} =\begin{bmatrix}{\mathrm{1}}\\{\mathrm{2}}\\{\mathrm{1}}\end{bmatrix}{k} \\ $$$$ \\ $$$${Case}\:{ii}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\lambda\:=\:\mathrm{3}\:\:\:{To}\:{substitite}\:\lambda\:{value}\:{in}\:\left[{A}−\lambda{I}\right]{X}\:=\:\mathrm{0} \\ $$$$\begin{bmatrix}{\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:−\mathrm{1}\:\:\:\:\:\:\mathrm{0}}\\{−\mathrm{1}\:\:\:\:\:\:\:−\mathrm{1}\:\:−\mathrm{1}}\\{\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:\:−\mathrm{1}\:\:\:\:\:\:\mathrm{0}}\end{bmatrix}\begin{bmatrix}{{x}_{\mathrm{1}} }\\{{x}_{\mathrm{2}} }\\{{x}_{\mathrm{3}} }\end{bmatrix}=\:\begin{bmatrix}{\mathrm{0}}\\{\mathrm{0}}\\{\mathrm{0}}\end{bmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left({R}_{\mathrm{1}} \Leftrightarrow{R}_{\mathrm{2}} \right) \\ $$$$\begin{bmatrix}{\:−\mathrm{1}\:\:\:\:\:−\mathrm{1}\:\:\:\:\:−\mathrm{1}}\\{\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:−\mathrm{1}\:\:\:\:\:\:\:\:\:\mathrm{0}}\\{\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:−\mathrm{1}\:\:\:\:\:\:\:\:\mathrm{0}}\end{bmatrix}\begin{bmatrix}{{x}_{\mathrm{1}} }\\{{x}_{\mathrm{2}} }\\{{x}_{\mathrm{3}} }\end{bmatrix}=\:\begin{bmatrix}{\mathrm{0}\:}\\{\mathrm{0}}\\{\mathrm{0}}\end{bmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{R}_{\mathrm{3}} \Rightarrow{R}_{\mathrm{3}} −{R}_{\mathrm{2}} \\ $$$$\begin{bmatrix}{\:\:−\mathrm{1}\:\:\:\:\:\:−\mathrm{1}\:\:\:−\mathrm{1}}\\{\:\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:−\mathrm{1}\:\:\:\:\:\:\:\mathrm{0}}\\{\:\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:\:\mathrm{0}}\end{bmatrix}\begin{bmatrix}{{x}_{\mathrm{1}} }\\{{x}_{\mathrm{2}} }\\{{x}\mathrm{3}}\end{bmatrix}\:=\:\begin{bmatrix}{\mathrm{0}}\\{\mathrm{0}}\\{\mathrm{0}}\end{bmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:{x}_{\mathrm{2}} =\:\mathrm{0} \\ $$$$\:\:\:\:−{x}_{\mathrm{1}} −{x}_{\mathrm{2}} −{x}_{\mathrm{3}\:} =\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:{x}_{\mathrm{3}} \:=\:{k} \\ $$$$\:\:\:\:\:\:\:\:{x}_{\mathrm{1}\:\:} =\:−{k}\: \\ $$$$\:\:\:\:\:\:\:\:\:{x}_{\mathrm{2}} =\:\begin{bmatrix}{−\mathrm{1}}\\{\:\:\:\:\mathrm{0}}\\{\:\:\:\:\mathrm{1}}\end{bmatrix}{k} \\ $$$${Case}\:{iii}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\lambda\:\:=\:\mathrm{4}\:\:{To}\:{substitute}\:{in}\:\:{equation} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left[\:{A}−\lambda{I}\:\right]{X}\:=\:\mathrm{0} \\ $$$$\begin{bmatrix}{\:\:−\mathrm{1}\:\:\:\:\:\:\:−\mathrm{1}\:\:\:\:\:\:\:\:\mathrm{0}}\\{\:\:−\mathrm{1}\:\:\:\:\:\:\:−\mathrm{2}\:\:\:−\mathrm{1}}\\{\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:−\mathrm{1}\:\:\:\:−\mathrm{1}}\end{bmatrix}\begin{bmatrix}{{x}_{\mathrm{1}} }\\{{x}_{\mathrm{2}} }\\{{x}_{\mathrm{3}} }\end{bmatrix}\:=\:\begin{bmatrix}{\mathrm{0}}\\{\mathrm{0}}\\{\mathrm{0}}\end{bmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{R}_{\mathrm{2}} \Rightarrow{R}_{\mathrm{2}} −{R}_{\mathrm{1}} \\ $$$$\begin{bmatrix}{\:\:−\mathrm{1}\:\:\:\:\:−\mathrm{1}\:\:\:\:\:\:\:\:\:\:\mathrm{0}}\\{\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:−\mathrm{1}\:\:\:\:\:−\mathrm{1}}\\{\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:−\mathrm{1}\:\:\:\:\:−\mathrm{1}}\end{bmatrix}\begin{bmatrix}{{x}_{\mathrm{1}} }\\{{x}_{\mathrm{2}} }\\{{x}_{\mathrm{3}} }\end{bmatrix}\:=\:\begin{bmatrix}{\mathrm{0}}\\{\mathrm{0}}\\{\mathrm{0}}\end{bmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{R}_{\mathrm{3}} \Rightarrow{R}_{\mathrm{3}} −{R}_{\mathrm{2}} \\ $$$$\begin{bmatrix}{\:\:−\mathrm{1}\:\:\:\:−\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\mathrm{0}}\\{\:\:\:\:\:\:\:\mathrm{0}\:\:\:\:−\mathrm{1}\:\:\:\:\:\:−\mathrm{1}}\\{\:\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:\:\:\:\mathrm{0}}\end{bmatrix}\begin{bmatrix}{{x}_{\mathrm{1}} }\\{{x}_{\mathrm{2}} }\\{{x}_{\mathrm{3}} }\end{bmatrix}\:=\:\begin{bmatrix}{\mathrm{0}}\\{\mathrm{0}}\\{\mathrm{0}}\end{bmatrix} \\ $$$$\:\:\:\:\:−{x}_{\mathrm{1}} −{x}_{\mathrm{2}\:} =\:\mathrm{0} \\ $$$$\:\:\:\:\:−{x}_{\mathrm{3}} −{x}_{\mathrm{3}} \:=\:\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:{x}_{\mathrm{3}\:} \:=\:\:\:\:{k} \\ $$$$\:\:\:\:\:\:\:\:{x}_{\mathrm{2}} \:=\:−{k} \\ $$$$\:\:\:\:\:\:\:\:{x}_{\mathrm{1}} \:=\:\:\:\:{k} \\ $$$$\:\:\:\:\:\:\:\:{x}_{\mathrm{3}\:\:} =\:\begin{bmatrix}{\:\:\:\:\mathrm{1}}\\{−\mathrm{1}}\\{\:\:\:\:\:\mathrm{1}\:\:}\end{bmatrix}{k} \\ $$$${we}\:{can}\:{observe}\:{that}\:{vectors}\:{are}\:{orthogonally}\:\:{mutually}\:\:{we}\:\:{normalize}\:{this}\:{vectors}\:{and}\:{obtain}. \\ $$$${e}_{\mathrm{1}} \:=\:\begin{bmatrix}{\frac{\mathrm{1}}{\:\sqrt{\mathrm{6}}}}\\{\frac{\mathrm{2}}{\:\sqrt{\mathrm{6}}}}\\{\frac{\mathrm{1}}{\:\sqrt{\mathrm{6}}}}\end{bmatrix}\:\:\:\:\:{e}_{\mathrm{2}} =\:\:\begin{bmatrix}{\frac{−\mathrm{1}}{\:\:\:\sqrt{\mathrm{2}}}}\\{\:\:\:\:\:\mathrm{0}}\\{\:\:\:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}}\end{bmatrix}\:\:\: \\ $$$$ \\ $$$${e}_{\mathrm{3}} \:=\begin{bmatrix}{\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}}\\{\frac{−\mathrm{1}}{\:\sqrt{\mathrm{3}}}}\\{\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}}\end{bmatrix} \\ $$$$\:\:\:\:\:\:\:\: \\ $$$${let}\:{P}\:=\:{the}\:{modal}\:{matrix}\:{in}\:{normalized}\:{form}\: \\ $$$${P}\:=\:\left[{e}_{\mathrm{1}\:\:} {e}_{\mathrm{2}} \:{e}_{\mathrm{3}} \right]\:=\begin{bmatrix}{\frac{\mathrm{1}}{\:\sqrt{\mathrm{6}}}\:\:\:\:\:\:\:\frac{−\mathrm{1}}{\:\sqrt{\mathrm{2}\:}}\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}}\\{\frac{\mathrm{2}}{\:\sqrt{\mathrm{6}\:}\:}\:\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:\:\:\:\frac{−\mathrm{1}}{\:\sqrt{\mathrm{3}}}}\\{\frac{\mathrm{1}}{\:\sqrt{\mathrm{6}}}\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\:\:\:\:\:\:\frac{−\mathrm{1}}{\:\sqrt{\mathrm{3}}}}\end{bmatrix} \\ $$$${Since}\:{p}\:{is}\:{a}\:{orthogonal}\:{matrix}\:{P}^{−\mathrm{1}\:} =\:\:{P}^{{T}} \\ $$$$\:{So}\:{D}\:=\:{P}^{{T}} {AP}\:\:\:\:{where}\:\:{D}\:\:{is}\:{diagnol}\:{matrix}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\begin{bmatrix}{\frac{\mathrm{1}}{\:\sqrt{\mathrm{6}}}\:\:\:\:\:\:\:\:\frac{−\mathrm{2}}{\:\sqrt{\mathrm{6}}}\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{6}}}}\\{\frac{−\mathrm{1}}{\:\sqrt{\mathrm{2}}}\:\:\:\:\:\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}}\\{\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}\:\:\:\:\:\:\:\:\frac{−\mathrm{1}}{\:\sqrt{\mathrm{3}}}\:\:\:\:\:\:\:\:\:\:\:\:\frac{−\mathrm{1}}{\:\sqrt{\mathrm{3}}}}\end{bmatrix}\begin{bmatrix}{\:\:\:\:\mathrm{3}\:\:\:\:\:\:\:−\mathrm{1}\:\:\:\:\:\:\:\:\:\mathrm{0}}\\{−\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}\:\:\:\:\:−\mathrm{1}}\\{\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:−\mathrm{1}\:\:\:\:\:\:\:\:\:\:\mathrm{3}}\end{bmatrix}\begin{bmatrix}{\frac{\mathrm{1}}{\:\sqrt{\mathrm{6}\:}}\:\:\:\:\:\:\:\:−\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}}\\{\frac{\mathrm{2}}{\:\sqrt{\mathrm{6}}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:\:\:\:\frac{−\mathrm{1}}{\:\sqrt{\mathrm{3}}}}\\{\frac{\mathrm{1}}{\:\sqrt{\mathrm{6}}}\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\:\:\:\:\:\:\:−\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}}\end{bmatrix} \\ $$$${D}\:\:=\:\begin{bmatrix}{\mathrm{1}\:\:\:\mathrm{0}\:\:\:\:\mathrm{0}\:}\\{\mathrm{0}\:\:\:\:\mathrm{3}\:\:\:\mathrm{0}}\\{\mathrm{0}\:\:\:\:\mathrm{0}\:\:\:\:\mathrm{4}}\end{bmatrix} \\ $$$${The}\:{quadratic}\:{form}\:{can}\:{be}\:{reduce}\:{to}\:{normal}\:{form}\:{Y}\:^{{T}} \:{DY} \\ $$$${Y}^{\:{T}} {DY}\:=\:\left[{y}_{\mathrm{1}} \:\:{y}_{\mathrm{2}\:} \:{y}_{\mathrm{3}} \right]\:\begin{bmatrix}{\mathrm{1}\:\:\:\:\mathrm{0}\:\:\:\:\:\mathrm{0}}\\{\mathrm{0}\:\:\:\:\mathrm{3}\:\:\:\:\:\mathrm{0}}\\{\mathrm{0}\:\:\:\:\mathrm{0}\:\:\:\:\mathrm{4}}\end{bmatrix}\:\begin{bmatrix}{{y}_{\mathrm{1}} }\\{{y}_{\mathrm{2}} }\\{{y}_{\mathrm{3}} }\end{bmatrix} \\ $$$$\:\:\:\:{y}_{\mathrm{1}} ^{\mathrm{2}} +\mathrm{3}{y}_{\mathrm{2}} ^{\mathrm{2}} +\mathrm{4}{y}_{\mathrm{3}} ^{\mathrm{2}\:} \:\:=\:\mathrm{0}\: \\ $$$$\:\:\:\:{Rank}\:\:\:\:\:\:\:\:\:\:=\:\mathrm{3}\:\:\:\:\:\:\:\:{Index}\:=\:\mathrm{3} \\ $$$$\:\:\:{Signature}=\:\mathrm{3}\:\:\:\:\:\:\:{Nature}={Positive}\:{definite} \\ $$$$\:{By}\:{orthogonal}\:{tfansformation}\:{X}\:=\:{PY} \\ $$$$\begin{bmatrix}{{x}_{\mathrm{1}} }\\{{x}_{\mathrm{2}} }\\{{x}_{\mathrm{3}} }\end{bmatrix}\:=\:\begin{bmatrix}{\frac{\mathrm{1}}{\:\sqrt{\mathrm{6}}}\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{−\mathrm{1}}{\:\sqrt{\mathrm{2}}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}}\\{\frac{\mathrm{2}}{\:\sqrt{\mathrm{6}}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{−\mathrm{1}}{\:\sqrt{\mathrm{3}}}}\\{\frac{\mathrm{1}}{\:\sqrt{\mathrm{6}\:\:}}\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}}\end{bmatrix}\begin{bmatrix}{{y}_{\mathrm{1}} }\\{{y}_{\mathrm{2}} }\\{{y}_{\mathrm{3}} }\end{bmatrix} \\ $$$${x}_{\mathrm{1}\:\:} =\:\frac{{y}_{\mathrm{1}} }{\:\sqrt{\mathrm{6}}}−\frac{{y}_{\mathrm{2}} }{\:\sqrt{\mathrm{2}}}+\frac{{y}_{\mathrm{3}} }{\:\sqrt{\mathrm{3}}}. \\ $$$${x}_{\mathrm{2}} =\:\frac{\mathrm{2}{y}_{\mathrm{1}} }{\:\sqrt{\mathrm{6}}}−\frac{{y}_{\mathrm{3}} }{\:\sqrt{\mathrm{3}.}} \\ $$$${x}_{\mathrm{3}} =\:\frac{{y}_{\mathrm{1}} }{\:\sqrt{\mathrm{6}}}+\frac{{y}_{\mathrm{2}} }{\:\sqrt{\mathrm{2}}}+\frac{{y}_{\mathrm{3}} }{\:\sqrt{\mathrm{3}.}} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$\: \\ $$$$\:\:\:\:\:\:\: \\ $$$$ \\ $$$${o} \\ $$