Question

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 .

Answer #1

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

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)...

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)...

v is an eigenvector with eigenvalue 5 for the invertible matrix
A. Is v an eigenvector for A^-2? Show why/why not.

Let
T : P(R) → P(R) be the linear map defined by T(p(x)) = xp′(x) (you
may take it for granted that T is linear). Show that for each λ ∈ Z
with λ ≥ 0, λ is an eigenvalue of T , and xλ is a corresponding
eigenvector.

The matrix A has an eigenvalue λ with an algebraic multiplicity
of 5 and a geometric multiplicity of 2. Does A have a generalised
eigenvector of rank 3 corresponding to λ? What about a generalised
eigenvector of rank 5?

Verify that u=[1,13]T is an eigenvector of the matrix
[[ -8,1],[-13,6]]. Find the corresponding eigenvalue lambda.

A system of diﬀerential equations having the form
t(~x)' = A ~x,
where A is a matrix with constant entries, is known as a
Cauchy-Euler system.
(a) Suppose λ is an eigenvalue of A and ~ v is an eigenvector
corresponding to λ. Show that the function
x(t) = t^λ (v)
is a solution to the Cauchy-Euler system t(x)'
= A(x).
(b) Solve the following Cauchy-Euler system:
t(x)' =
3
-2
2
-2
(x)
(t > 0)

Suppose A is a diagonalisable matrix and let k ≥ 1 be an
integer. Show that each eigenvector of A is an eigenvector of
Ak and conclude that Ak is diagonalisable

Use the power method to determine the highest eigenvalue and
corresponding eigenvector for
[ (2-λ) 8 10; 8 (4-λ) 5; 10 5 (7-λ)]
(By hand with handy formulas please)

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