Question

One point per regular blank unless otherwise specified, two points per blank matrix. M=CE determinant of M= trace of M= (2 points) The characteristic polynomial PM(x) = det (xl – M) = Find the two eigenvalues of M, the roots of the characteristic polynomial. 21 = – 12 = M can be decomposed into two idempotent matrices Ej and E2, with the following properties. El + E2 = 1 1E1 + 12E2 = M E Ez = [0] E2 = In class, we learned that any real valued function can be performed on the matrix M by using the eigenvalues, f(M) = f(11)E1 + f(12)E2. Consider square roots. The principal square root of 4 is 2, but (-2)2=4 as well. Using this principle, there are four different matrices that can be considered VM. Find all four. first version of VM second version of M= first version of Vm = L ehird version of vm=[ second version of v = [ fourth version of via = [ ] ] third version of M = fourth version of VM = Find the eigenvalues of M². 11 = 22 =

Answer 1

Given m= 15 ] det (m) = 25-16 = 9 To(m) = 5+5 = 10 Characteristic polynomial Pmln) = det (XI-M) = 22- TrIM)x+ de+(M) 22. 10x + 9 Equate it to zero we get 1x²_100 +9=0l charactens se Equation → x?94-X +9=0 = 217-9)-1(8-9)=0 - (2-1) (3-9) = 0 =) . 221, 7-9 1 = 1 and 11 = 9] Now in can be decomposed into two Idempotent matrizes E, and E Sucis that E,+E, = I, 1, 6, 71, E, EM and E, €2=10]

Nole: Necessam ondition for ax? mahx to be. Idem potent is that either it is diagonal or its trace equais / so Any matrix of the form lab with a²+bc = a is (ci-a) dem polent. ca, bz1 Now let E, = [ a b ] & E,= [a b ) Now E, TE, = I =) (914a2 bitha 7 1 4+2 2-4,-921 - 9179,- bitb,=0 2-9,792 = 1 etc, co - A, E, + 12 E₂ = M = 4(9:6,]+91 97, 7+(35) 9179a₂=5 5,+962 = 8 C1 + 942 = 2 1-a, +9 (1-92) = 5

Now - we have aita=1 & 9,+992=5 = a + 9 = 5 - - 9 + 92=1 89 =4 = | 9 = 1/2) 19,= Also b6 + 9b₂=8 51+ b2 so [b2=1 ² 862 = 8 → 5,= -1 Also 9+96=2 Atų =0 - - - 24 = 2 = {2 = 14 - [ C= -1/4

Noы Слин ЕЕ, = Ce] - І, Јв, ) = ( ; - 1- гоо] 1- 4+, -,+ , ) Loo), / we are loo, Treuf Also eigen rauses of m are 1,9 а - €i реn vаще, м