Question

A square matrix *A* is said to be idempotent if
*A*^{2} = *A*. Let *A* be an
idempotent matrix.

- Show that
*I**−**A*is also idempotent. - Show that if
*A*is invertible, then*A*=*I*. - Show that the only possible eigenvalues of
*A*are 0 and 1.(Hint: Supposeis an eigenvector with associated eigenvalue*x**λ*and then multiplyon the left by*x**A*twice.) - Let
*W*= col(*A*). Show that*T*(_{A}) = proj*x**W*and*x**T*_{I}_{−}(_{A}) = proj*x*⊥_{W}.*x*

Answer #1

I hope you will get it

A square matrix A is said to be idempotent if A2 = A. Let A be
an idempotent matrix.
Show that I − A is also idempotent.
Show that if A is invertible, then A = I.
Show that the only possible eigenvalues of A are 0 and 1.(Hint:
Suppose x is an eigenvector with associated eigenvalue λ and then
multiply x on the left by A twice.) Let W = col(A).
Show that TA(x) = projW x and TI−A(x)...

Let A be an n × n matrix and let x be an eigenvector of A
corresponding to the eigenvalue λ . Show that for any positive
integer m, x is an eigenvector of Am corresponding to the
eigenvalue λ m .

Let
P be a stochastic matrix. Show that λ=1 is an eigenvalue of P. What
is the associated eigenvector?

Problem 3.2
Let H ∈ Rn×n be symmetric and idempotent, hence a projection
matrix. Let x ∼ N(0,In). (a) Let σ > 0 be a positive number.
Find the distribution of σx. (b) Let u = Hx and v = (I −H)x and ﬁnd
the joint distribution of (u,v). 1 (c) Someone claims that u and v
are independent. Is that true? (d) Let µ ∈ Im(H). Show that Hµ = µ.
(e) Assume that 1 ∈ Im(H) and ﬁnd...

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