Question

Please solve in details.
Show that, u, v, w are orthonormal eigenvectors of matrix M, corresponding to eigenvalues 1, 12, 13 respectively, and L is a square matrix whose columns are u, v, w , then (D = Ll' (M L ) is a diagonal matrix.

Answer 1

Let M be an nxn matrix. and 1, 2, & be three eigen values of M with eigen vectors u=1 U2 W=W2 Wn] Consider Tun Un wnd 4x3 and I am uz... un 7 Wz ... wn 3.n also given that u v w are ortho normal ie, ut v = v^ u = win = utw = VT w = WTV=O. and uTo = vT u = WT W = 1.

Now consider [D] = LTML cleardly [D] is a 3x3 - matrix. So, Define Ti B3 B3 as. 2 T(x) = Dx is a Linear map, with matrix of T with respect to standard basis {eres, es} is [0]. where, e,= (1,0,0), ea=(0-10), e3= (0,0,1). Consider. Tie) = De ELTML Now consider wil 4(!) = (x, na U2 wa lol = 42 Un in un WX X

Now LT Mie = LTM u - LT 1 U Csince, o is the eigen vectoro Cororesponds to » Mu=2,0). * LT MLE, = ? LT u. Considero LT U = Ua... Un Tun wi wa... wn]: 1) (using the Past and ng the fact that uw w are Choomal) oothonooma => LTM LE, = się

( Tle,) = 1, e, = ), e, toe toe Similarly Tleal = 2 la = 0e, tilat ol Tles) = q es = 0e, tola + y es 2 [T] = [1,007 J3x3 I the matrix of T with Standard basis] respect to the Since [T] = [D] we get. D=007 los 0 7 D = LTML is diagonal matrix.