Question

1. The symmetric matrix [4 1 1 1 A = 4-1 1 -1 4 -1 4 -1 has eigenvalues A = 1 (with algebraic multiplicity 1) and A 5 (with algebraic multiplicity 3). a) Find bases for the eigenspaces E(1) and E(5). b) Apply the Gram-Schmidt process to your basis for E(5) to find an onal basis for E (5) orthog- (c) Hence write down an that QT AQ = D. orthogonal matrix Q and a diagonal matrix D such (d) Write down the quadratic form associated with the matrix A, and write down an equivalent quadratic form in new variables that contains no cross terms.

Answer 1

IThe iuen matix i's A 4-1 -I I- -1 Its eigenlues cre (ith algebraic mult pligty 5 (with alsebraic multiplichy 3) and A) let V cisenvectr cospondin to ge cen Then CA-I) 4-1 4-1 -I 4- 4- T The augnented matri is -1 1 R R 1 3 1 - I -1 - I -1 3 RR3R - 8 C 4 C 4 RR4 C -4 - & -1 3 -l 2. 1 - C - I 1 R4R8R 4

C > ,z-t,t, puthiny t .V, t ELU= A bas for to A .cijuect anpondig he an A-SI) hen 4-S 4-S 4-S -1 4-J 1 - 1 7he aueuteod at 1s R R -1 RAR - I - +St A bass for ElS)

(6) Gram-Schmidr proiess to fiel an oosonal basis for Es) A basis for Els) , whare be he comspauliy ohogonal basis for ElS) Taen 'nM W,w CIC)+C0C)+ C) () + (0)(9) C 12 1/2 COC+ +( (U() wCD+COCU+ (20CIJ() )c2 (1 7 /2 ELS) Ah orthogonal kas for cc) Wutey0 2 11W

w 141 l1필1 1일 oruogaal metrix / .1 An TAQ D where is uch tha Quadratic fom associated win ati A is (d) 4 AR=, 4 - 4 4x 2%- 27,4, 22A4 +2 2 , u, + 2 U , QTAQ= D A QDQ AR)D AR TD ( A D where ie. The cores ondi equivaleut 9adat for n no crosses 9'D7 0 o S o O S 1-