Suppose A is the matrix for T R3 rightarrow R3 relative to t

Suppose A is the matrix for T: R^3 rightarrow R^3 relative to the standard basis. Find the diagonal matrix A\' for T relative to the basis B\'. A = [5/2 1/2 1 -1/2 3/2 -1 1/2 -1/2 2], B\'={(1, 1, -1), (1, -1, 1), (-1, 1, 1)} A\' =

Solution

A is given ,

then T(x,y,z) = [5/2x +1/2y +z , -1/2x +3/2y -1z, 1/2x -1/2y+2z ]

The eigen values are 1,2,3

eigen vectors are (-1,1,1) ,(-1,-1,1), (1,-1,1)

now T (-1,1,1) = 1( (-1,1,1)

T(-1,-1,1) = 2(-1,-1,1)

T(1,-1,1) = 3(1,-1,1)

therefore matrix of T with respect to eigen vector will be a diagonal matrix with element as 1,2,3