Question

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) = projW ⊥ x.

Answer #1

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...

(3 pts) Let A be a square n × n matrix whose rows are
orthogonal. Prove that the columns of A are also orthogonal.
Hint: The orthogonality of rows is equivalent to AAT
= I ⇒ ATAAT = AT

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