Question

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

Answer #1

Let A be a matrix with an eigenvalue λ that has an algebraic
multiplicity of k, but a geometric multiplicity of p < k, i.e.
there are p linearly independent generalised eigenvectors of rank 1
associated with the eigenvalue λ, equivalently, the eigenspace of λ
has a dimension of p. Show that the generalised eigenspace of rank
2 has at most dimension 2p.

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 .

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?

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)...

Suppose A is an orthogonal matrix. Show that |λ| = 1 for all
eigen-
values λ. (Hint: start off with an eigenvector and dot-product
it with itself.
Then cleverly insert A and At into the dot-product.)
b) Suppose P is an orthogonal projection. Show that the only
possible
eigenvalues are 0 and 1. (Hint: start off with an eigenvector
and write down
the definition. Then apply P to both sides.)
An n×n matrix B is symmetric if B = Bt....

We say the a matrix A is similar to a matrix B if there is some
invertible matrix P so that B=P^-1 AP.
Show that if A and B are similar matrices and b is an eigenvalue
for B, then b is also an eigenvalue for A. How would an eigenvector
for B associated with b compare to an eigenvector for A?

Assume A is an invertible matrix
1. prove that 0 is not an eigenvalue of A
2. assume λ is an eigenvalue of A. Show that λ^(-1) is an
eigenvalue of A^(-1)

Let ? be an eigenvalue of the ? × ? matrix A. Prove that ? + 1
is an eigenvalue of the matrix ? + ?? .

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

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