Let A be an n x n matrix satisfying A2=A
(idempotent). Find all eigenvalues and eigenvectors of A.
I know that the eigenvalues are 0 and 1 -- I do not know how to
find the eigenvectors.
Let A be a diagonalizable n × n matrix and let P be an
invertible n × n matrix such that B = P−1AP is the diagonal form of
A. Prove that Ak = PBkP−1, where k is a positive integer. Use the
result above to find the indicated power of A. A = 6 0 −4 7 −1 −4 6
0 −4 , A5 A5 =
Let A be a diagonalizable n × n
matrix and let P be an invertible n × n
matrix such that
B = P−1AP
is the diagonal form of A. Prove that
Ak = PBkP−1,
where k is a positive integer.
Use the result above to find A5
A =
4
0
−4
5
−1
−4
6
0
−6
Let A be some m*n matrix. Consider the set S = {z : Az = 0}.
First show that this is a vector space. Now show that n = p+q where
p = rank(A) and q = dim(S). Here is how to do it. Let the vectors
x1, . . . , xp be such that Ax1, .
. . ,Axp form a basis of the column space of A (thus
each x can be chosen to be some unit...
Let A be an m × n matrix and B be an m × p matrix. Let C =[A |
B] be an m×(n + p) matrix.
(a) Show that R(C) = R(A) + R(B), where R(·) denotes the range
of a matrix.
(b) Show that rank(C) = rank(A) + rank(B)−dim(R(A)∩R(B)).
Let v1 be an eigenvector of an n×n matrix A corresponding to λ1,
and let v2, v3 be two linearly independent eigenvectors of A
corresponding to λ2, where λ1 is not equal to λ2. Show that v1, v2,
v3 are linearly independent.
Let 3x3 matrix A = -3 0 -4
0 5 0
-4 0 3
a) Find the eigenvalues of A and list their multiplicities.
b) Find a basis, Bi, for each eigenspace, E(i).
c) If possible, diagonalise matrix A. (i.e find matrices P and D
such that Pinv AP = D is diagonal).
For each n ∈ N, let fn : [0, 1] → [0, 1] be defined
by fn(x) = 0, x > 1/n and fn(x) = 1−nx if
0 ≤ x ≤1/n.
The collection {fn(x) : n ∈ N} converges to a point,
call it f(x) for each x ∈ [0, 1]. Show whether {fn(x) :
n ∈ N}
converges to f uniformly or not.
Let A be a real n × n matrix, and suppose that every leading
principal submatrix ofA of order k < n is nonsingular. Show that
A has an LU-factorisation.