Lecture 14

Question 1

Range(A)

Answer 1

{y = Ax | \forall x}

Question 2

Null(A)

Answer 2

{x | Ax = 0}

Question 3

Eigenvector/eigenvalue

Answer 3

x ≠ 0 eigenvector of A if there’s Ax = λx, λ eigenvalue. Usually normalized eigenvectors ||x||=1 (because αx is also eigenvector)

Question 4

Find eigenvalues

Answer 4

Ax = λx
(A-λI)x=0
p(λ)=det(A-λI)=0 (characteristic polynomial of degree n)
Roots of p(λ) are eigenvalues
Only analytical solutions for n<=4, otherwise approx num –LU would give exact solution if no floating point error.

Question 5

Eigenvalues properties

Answer 5

Can have zero/complex/negative values. Eigenvalues can have a multiplicity.

Question 6

A =
1  e
e  1
where e=1e-10. Select the correct statement:
A) λ1 = 1
B) λ1 = 0
C) λ1 slightly>1
D) λ1 slightly>0

Answer 6

C

Question 7

Diagonalizable matrices

Answer 7

A nxn with n linearly independent eigenvectors u.

AU = UD, A=UDU-1 similarity transformation of A (D with λi ordered, U non singular)

Question 8

Power iteration

Answer 8

xk = λ1^k[α1.u1 + α2(λ2/λ1)^k.u2 + … + αn(λn/λ1)^k.un

if k → ∞, xk → α1.λ1^k.u1 multiple of u1.

Question 9

Rayleigh coefficient

Answer 9

λ= xTAx / xTx

Question 10

Power iteration normalized

Answer 10

yk = A.x_{k-1}
xk = yk/||yk||
avoids possible overflow or underflow

Question 11

Power iteration, what if α1=0? (x0 has no component in the dominent eigenvector u1)

Answer 11

theory: xk→λ2^k α2 u2 (converges to u2 instead) BUT usually FP operations will introduce such component and α1≠0.

Question 12

Power iteration, what if |λ1|=|λ2|?

Answer 12

Does not work, xk→λ1^k α1 u1 + λ2^k α2 u2

• if sign are the same, method will converge to the correct eigenvalue
• otherwise, no convergence

Question 13

Power iteration error

Answer 13

||ek|| = O(|λ2/λ1|^k)

Question 14

Power iteration convergence rate

Answer 14

linear, ||ek+1||//||ek|| = |λ2|/|λ1|

Question 15

Similarity transformation changes eigenvalues/eigenvectors?

Answer 15

changes eigenvectors but not eigenvalues

Question 16

Transformations that does not change the eigenvectors of a matrix but changes eigenvalues?

Answer 16

• Taking any power of the matrix
• Inverting the matrix
• Shifting by a scalar multiple of the identity matrix

Question 17

λA1=3, λA2=-1, B=((A-2I)^-1)^2, λB1,λB2=?

Answer 17

1, 0.11

Question 18

How many power iterations to reduce the error by a factor of >1e-3?

Answer 18

||ek|| = |λ2/λ1|^k ||e0||

so k> log(1e-3)/log(λ2/λ1)

Question 19

Cost power iteration

Answer 19

O(n^2) per iteration