Question

the space spanned by H () and) 14. Suppose P projects 4-dimension al vectors ont 2,3) and (1.1,-1.-1). Ple ase answer the following questions about P (a) What are the eigenvalues of P? (b) Is P diagon aliz able? Justify your answer c)Find a maximal set of line arly independent eigenvectors of P and indicate which eigenvalues they correspond to. (d) Suppose you were going to construct the 4 x 4 matrix corresponding to P via P- A(ATA) AT What is A?

Answer 1

P (matrix (1/2 1/2 1/2 1/2)) = matrix (1/2 1/2 1/2 1/2)) p (matrix (1/2 1/2 -1/2 -1/2)) = matrix (1/2 1/2 -1/2 -1/2) eigen value is 1, corresponds eigen vectors matrix (1/2 1/2 1/2 1/2) matrix (1/2 1/2 -1/2 -1/2) matrix (1/2 1/2 1/2 1/2) matrix (a b c d) = 0;matrix (1/2 1/2 -1/2 -1/2) matrix (a b c d) = 0: a + b + c + d = 0, a + b - c - d = 0: a + b = 0: b = -a: d = -c: matrix (a -a c -c) = a matrix (1 -1 0 0) + c matrix (0 0 1 -1) p matrix (1 -1 0 0) = matrix (0 0 0 0): p matrix (0 0 1 -1) = matrix (0 0 0 0) \r\n it is diognalizble since eigen values 0, 0 two eigen vectors, eigen value 1