and eigenvalue λ1 = 3 and associated eigenvector v(1) = (1, 0,
1)t . Find the second dominant eigenvalue λ2 (or the approximation
to λ2) by the Wielandt Dflation method
Let A = [ 0 2 0
1 0 2
0 1 0 ] .
(a) Find the eigenvalues of A and bases of the corresponding
eigenspaces.
(b) Which of the eigenspaces is a line through the origin? Write
down two vectors parallel to this line.
(c) Find a plane W ⊂ R 3 such that for any w ∈ W one has Aw ∈ W
, or explain why such a plain does not exist.
(d) Write down explicitly...
Let A = 0 2 0
1 0 2
0 1 0 .
(a) Find the eigenvalues of A and bases of the corresponding
eigenspaces.
(b) Which of the eigenspaces is a line through the origin? Write
down two vectors parallel to this line.
(c) Find a plane W ⊂ R 3 such that for any w ∈ W one has Aw ∈ W
, or explain why such a plain does not exist.
(d) Write down explicitly a diagonalizing...
Let A = 0 2 0
1 0 2
0 1 0 .
(a) Find the eigenvalues of A and bases of the corresponding
eigenspaces.
(b) Which of the eigenspaces is a line through the origin? Write
down two vectors parallel to this line.
(c) Find a plane W ⊂ R 3 such that for any w ∈ W one has Aw ∈ W
, or explain why such a plain does not exist.
(d) Write down explicitly a diagonalizing...
Let X = [1, 0, 2, 0]tand Y = [1, −1, 0, 2]t.
(a) Find a system of two equations in four unknowns whose
solution set is spanned by X and Y.
(b) Find a system of three equations in four unknowns whose
solution set is spanned by X and Y.
(c) Find a system of four equations in four unknowns that has
the set of vectors of the form Z + aX + bY as its general solution
where...
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.
Which of the following statements is/are true?
1) If a matrix has 0 as an eigenvalue, then it is not
invertible.
2) A matrix with its entries as real numbers cannot have a
non-real eigenvalue.
3) Any nonzero vector will serve as an eigenvector for the
identity matrix.
Let f(x, y) be a function such that f(0, 0) = 1, f(0, 1) = 2,
f(1, 0) = 3, f(1, 1) = 5, f(2, 0) = 5, f(2, 1) = 10. Determine the
Lagrange interpolation F(x, y) that interpolates the above data.
Use Lagrangian bi-variate interpolation to solve this and also show
the working steps.
1. If (1, -1) is an eigenvector of A with associated
eigenvalue -2, and (1, 1) is an eigenvector of A with
associated eigenvalue 4, then what the entries of A ,a11 , a12,
a21 and a22 ?
2. If A has a repeated eigenvalue, the A definitely isn't
diagonalizable. (True or False)