Primary decomposition / minimum polynomial

Question 1

What is the smallest possible subspace containing U and W

Answer 1

U+W={u+w : u∈U , w∈W}

Question 2

Direct sum

Answer 2

U+W=U⊕W is a direct sum if

(a) whenever u∈U w∈W and u+w=0 then u=w=0
(b) if u∈U∩W then u=0 i.e. the intersection is {0}

Question 3

How to find a basis of U+W (given a basis of U and basis of W)

Answer 3

1. take one of the bases
2. select elements from the other basis and put them in the new basis if they are linearly independent. If not, then discard.

Question 4

U+W is a direct sum i.e. U+W=U⊕W when the basis of U+W is…

Answer 4

both the complete bases of U and W

Question 5

If V has dimension a+b and dim(U)=a and dim(W)=b then…

Answer 5

a basis exits such that U+W=U⊕W

Question 6

Proposition 11.4. bases of a direct sum.

Answer 6

(a) if e₁,…,eₖ is a basis of U and f₁,…,fₗ is a basis of W and U+W is a direct sum then e₁,…,eₖ,f₁,…,fₗ is a basis of U+W=U⊕W
(b) if e₁,…,eₖ,f₁,…,fₗ is linearly independent and U=span{e₁,…,eₖ} W = span{f₁,…,fₗ} then U+W=U⊕W

Question 7

Theorem 11.5 . direct sums of eigenspaces.

Answer 7

suppose m(x)=p(x)q(x)  where p and q have no common factors. A is a matrix. Define 
U=ker(m(A)) => Up = ker(p(A)), Uq = ker(q(A)) Then 
U = Up ⊕ Uq

Question 8

Theorem 11.6 . Primary Decomposition Theorem.

Answer 8

Suppose the polynomial of A factorises fully (the characteristic polynomial) with distinct eigenvalues. Then we can define the subspaces
Uᵢ = ker(( λᵢI - A)ᵉᶦ) ( the generalised eigenspace)
Then V is the direct sum of all the Uᵢs

Question 9

Ker(B) = ker(-B) true or false?

Answer 9

true

Question 10

Corollary 11.7. (of primary decomposition theorem).

Answer 10

A is diagonisable if and only if the characteristic polynomial factorises fully.

Question 11

How to find the generalised eigenvectors

Answer 11

take the image

Question 12

monic

Answer 12

A polynomial is monic if its leading coefficiant is 1

Question 13

minimum polynomial

Answer 13

the unique monic polynomial of least degree s.t. m(A)=0

Question 14

proposition 12.2. minimum polynomial and characteristic polynomial.

Answer 14

The minimum polynomial is well defined and divides the characteristic polynomial.

Question 15

proposition 12.3. minimum polynomial of an eigenvalue.

Answer 15

m(λ)=0

Question 16

How to calculate the minimum polynomial

Answer 16

1. calculate the characteristic polynomial
2. factorise the characteristic polynomial/calculate eigenvalues.
3. if the characteristic polynomial is monic and has no repeated roots than this is m(x)
4. otherwise make it monic and for each power of each repeated root calculate (A-aI)ᵏ(rest of characteristic polynomial). choose the k when this is all equal to 0.

Question 17

proposition 12.7. diagonisable when (eigenvectors)

Answer 17

A is diagonisable if there is a basis of eigenvectors of A

Question 18

proposition 12.9 diagonisable when (minimum polynomial)

Answer 18

A is diagoniable if m(x) has no repeated roots