Question

Let y = x Ax be a quadratic form where x ER" and A ERNXN. What ду is in terms of x and A? дх

Please explain the answer in detail and in good hand writing! Thanks a lot!

Answer 1

Given, Y=XTAX where XEIR and At Rnyn let, A= 4,2. 913 ain 021 A22 223 aan € Rnxn an anz ang ann nym and X = X ERD xn Inxi Now AX= an 012 043 ain Azi 922 aan ani ana anz ann xn nxn aixt 012 ² + ... tanan A21 4+ a22²₂ + + aanxn amt t am & ... + annan nxi. Now, TAX = AX=[ anat a 2 + ... + aman 1 X ... 2 ixn 9214 + O2 2% + ... + ankw and 4t and X₂ + ... + annan ny

xlarix + a2 + ... + amxn) + %( 9 x + 922} + ... tanta) 4 ann) t... & Xnan 4+ ane & xan & An, X t ... & ani tn) + H (Q2 2 + 0,2 % t... & ana Xon) + + xn (anur an de + & annan) xi aij xj i- j=1 512;; xi?+ Li Qiz ay) jti with respect to the Take the partial derivative element &K, ay X auxi²+ I x aij dj jti = 20kk + { di Ojk & [ axjxj j#K j #K First term comes from are which is quadratic in 4, and and 3rd term comes which are linear in a

+ EaRj lj jal We move one Okkkk 12 into each of the sum of 2nd and 3rd term geven in note t) (This explanation Now, ay ay 3% sjaj, + & a,; J-1 ay jayz + aj X sy saint & anity ajh yaji SWH SWH az IMs Ms Mo Ang ty X'ain angxi ja Now 21 ATX = ani 34 212 a, 2 Ana ETEL Xn an aan ann nX1 nin a 14 tag] Y₂ + ... + and an 01221 + a22 X₂ + ... + anem xuj2 19 ansu + Qan X₂ + ... + annan xajn

2) 11 122, 2, 3 .- AX= + and (21X1+ 02 , + + + + (ทใ ay at ang X₂ + ... + annan aj xj I Quy ay S j Anja from 2. ATX + AX - 2Y (AT+AX

Note :- 1) We have y = x (01124 + 2X₂ + ... + ax ax +...+ ankn) + X (Ag x + a22X₂ + ... tak kx + ... + Anan) + + 241 anixt apa Z + ... + axxx t.. taknan) + tan (Ama & And he + ... & And X4 + ... + aman). partial derivative W.r. to k. dy 2xx 4 ar & ankt... ph Arkt.it Xn Ank + (axi & + Ak2 let... + ak akx + ... + X, akn). - 2) If x-ajk + 2 anizo Ja ja f is a scalar function and x=(4, ... ,xn) then 24