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A Bruhat type decomposition of the set of conditioned invariant subspaces

Jochen Trumpf

Jochen.Trumpf@anu.edu.au

Department of Systems Engineering Research School of Information Sciences and Engineering The Australian National University and National ICT Australia Ltd.

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.1/22

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A Bruhat type decomposition of the set of conditioned invariant subspaces – p.2/22

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Brunovsky form

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.2/22

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Brunovsky form restrictions

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.2/22

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Brunovsky form restrictions restriction indices

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.2/22

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Brunovsky form restrictions restriction indices image representations

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.2/22

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Brunovsky form restrictions restriction indices image representations tightness

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restricted output injection

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.3/22

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restricted output injection Kronecker form

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.3/22

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restricted output injection Kronecker form restrictions

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.3/22

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restricted output injection Kronecker form restrictions image representations

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.3/22

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restricted output injection Kronecker form restrictions image representations generalised flag manifolds

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.3/22

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restricted output injection Kronecker form restrictions image representations generalised flag manifolds the retraction map

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.3/22

• utput injection

Given a pair of matrices (C, A) ∈ Rp×n × Rn×n the group action

(T, J, S), (C, A) → (SCT −1, T(A − JC)T −1)

where J ∈ Rn×p is arbitrary and T ∈ Rn×n and

S ∈ Rp×p are invertible is called the output

injection equivalence action.

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.4/22

• utput injection

Given a pair of matrices (C, A) ∈ Rp×n × Rn×n the group action

(T, J, S), (C, A) → (SCT −1, T(A − JC)T −1)

where J ∈ Rn×p is arbitrary and T ∈ Rn×n and

S ∈ Rp×p are invertible is called the output

injection equivalence action. Its corresponding normal form is the so-called Brunovsky normal form. In the case where (C, A) is observable it looks like this.

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.4/22

Brunovsky form

C =

    

0... 01

...

0... 01

µ1 . . . µp      and

A =

            

1...

......

1

...

1...

......

1

• µ1

. . .

• µp

            

where µ1 ≥ ... ≥ µp ≥ 0 are the observability indices of ther pair (C, A) which form a complete set of invariants.

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.5/22

restrictions

A conditioned invariant or (C, A)-invariant subspace is a subspace V ⊂ Rn for which there exists an output injection J such that

(A − JC)V ⊂ V

Note that this concept is invariant under output injection equivalence.

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.6/22

restrictions

A conditioned invariant or (C, A)-invariant subspace is a subspace V ⊂ Rn for which there exists an output injection J such that

(A − JC)V ⊂ V

Note that this concept is invariant under output injection equivalence. There is a natural notion of restriction of the pair

(C, A) to V as follows.

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.6/22

restrictions

Rn/V

˜ A

• Rn/V

˜ C

• Rp/C(V)

Rn A − JC

• Rn

C

• Rp
• V

¯ A

• V

¯ C

• C(V)
• A Bruhat type decomposition of the set of conditioned invariant subspaces – p.7/22

restriction indices

It can be shown that all matrix representations

( ¯ C, ¯ A) of restrictions are output injection

equivalent and that all pairs that are output injection equivalent to a given matrix representation of a restriction are in fact such.

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.8/22

restriction indices

It can be shown that all matrix representations

( ¯ C, ¯ A) of restrictions are output injection

equivalent and that all pairs that are output injection equivalent to a given matrix representation of a restriction are in fact such. Hence there is a matrix representation of a restriction in Brunovsky normal form. Its indices

(λ1, . . . , λp) are called the restriction indices of (C, A) with respect to V. It is λi ≤ µi for i = 1, . . . , p.

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.8/22

image representations

Theorem [FPP]: The codimension q (C, A)-invariant subspaces are precisely the images of full rank matrices

Z ∈ Rn×(n−q) which fullfill

(1) AZ = Z ¯

A + AZ ¯ C⊤ ¯ C

(2) CZ = CZ ¯

C⊤ ¯ C

(3) CZ ¯

C⊤ has full rank

where ( ¯

C, ¯ A) is the matrix representation of a

restriction in Brunovsky form.

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.9/22

image representations

The set of all these Zs is called M(λ, µ), we denote Γ(λ) = M(λ, λ) which is a group. In fact

Im Z1 = Im Z2 for Z1, Z2 ∈ M(λ, µ) if and only if Z2 = Z1S−1 for S ∈ Γ(λ).

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.10/22

image representations

The set of all these Zs is called M(λ, µ), we denote Γ(λ) = M(λ, λ) which is a group. In fact

Im Z1 = Im Z2 for Z1, Z2 ∈ M(λ, µ) if and only if Z2 = Z1S−1 for S ∈ Γ(λ).

Hence the Brunovsky stratum of all codimension

q conditioned invariant subspaces with fixed

restriction indices λ is a smooth manifold diffeomorphic to M(λ, µ)/Γ(λ).

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.10/22

image representations

The set of all these Zs is called M(λ, µ), we denote Γ(λ) = M(λ, λ) which is a group. In fact

Im Z1 = Im Z2 for Z1, Z2 ∈ M(λ, µ) if and only if Z2 = Z1S−1 for S ∈ Γ(λ).

Hence the Brunovsky stratum of all codimension

q conditioned invariant subspaces with fixed

restriction indices λ is a smooth manifold diffeomorphic to M(λ, µ)/Γ(λ). Equations (1) and (2) can be solved explicitely.

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.10/22

image representations

Z = (Zij), Zij =

                                                        

z1 ... z2

... ... . . . . . . ... ...

zµi−λj

...

z1 zµi−λj+1 ... z2

... ... . . . . . . ... ...

zµi−λj ... zµi−λj+1

             

, λj ≤ µi

0 ... 0

. . . . . .

0 ... 0

• , λj > µi

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.11/22

tightness

A (C, A)-invariant subspace V is called tight if

V + Ker C = Rn or, equivalently, if rk ¯ C = p, i.e. λp > 0.

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.12/22

tightness

A (C, A)-invariant subspace V is called tight if

V + Ker C = Rn or, equivalently, if rk ¯ C = p, i.e. λp > 0.

Then CZ ¯

C⊤ is invertible. We will consider only

this case.

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.12/22

restricted output injection

The same construction can be repeated starting from a slightly different group action

(T, J, U), (C, A) → (UCT −1, T(A − JC)T −1)

where U is unipotent lower triangular.

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.13/22

restricted output injection

The same construction can be repeated starting from a slightly different group action

(T, J, U), (C, A) → (UCT −1, T(A − JC)T −1)

where U is unipotent lower triangular. The normal form (Kronecker form) looks very similar.

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.13/22

Kronecker form

C =

    

0... 01

...

0... 01

K1 . . . Kp      and

A =

            

1...

......

1

...

1...

......

1

• K1

. . .

• Kp

            

but the Kronecker indices (K1, . . . , Kp) are not neccessarily ordered. They are a permutation of the observability indices (µ1, . . . , µp).

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.14/22

restrictions

Again we can look at restrictions of the pair

(C, A) to a (C, A)-invariant subspace V and define

the restricted Kronecker indices (k1, . . . , kp) which are a permutation of the restriction indices

(λ1, . . . , λp). It is ki ≤ µi for i = 1, . . . , p.

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.15/22

restrictions

Again we can look at restrictions of the pair

(C, A) to a (C, A)-invariant subspace V and define

the restricted Kronecker indices (k1, . . . , kp) which are a permutation of the restriction indices

(λ1, . . . , λp). It is ki ≤ µi for i = 1, . . . , p.

Again we get a characterisation of all bases of

(C, A)-invariant subspaces.

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.15/22

image representations

Theorem: The codimension q tight (C, A)-invariant subspaces are precisely the images of full rank matrices Z ∈ Rn×(n−q) which fullfill (1) AZ = Z ¯

A + AZ ¯ C⊤ ¯ C

(2) CZ = CZ ¯

C⊤ ¯ C

(3) CZ ¯

C⊤ is unipotent

where ( ¯

C, ¯ A) is the matrix representation of a

restriction in Kronecker form.

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.16/22

image representations

The set of all these Zs is called M(k, µ), we denote Γ(k) = M(k, k) which is a group. In fact

Im Z1 = Im Z2 for Z1, Z2 ∈ M(k, µ) if and only if Z2 = Z1S−1 for S ∈ Γ(k).

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.17/22

image representations

The set of all these Zs is called M(k, µ), we denote Γ(k) = M(k, k) which is a group. In fact

Im Z1 = Im Z2 for Z1, Z2 ∈ M(k, µ) if and only if Z2 = Z1S−1 for S ∈ Γ(k).

Hence the Kronecker stratum of all codimension

q conditioned invariant subspaces with fixed

restricted Kronecker indices k is diffeomorphic to

M(k, µ)/Γ(k) which is diffeomorphic to an RN.

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.17/22

image representations

The set of all these Zs is called M(k, µ), we denote Γ(k) = M(k, k) which is a group. In fact

Im Z1 = Im Z2 for Z1, Z2 ∈ M(k, µ) if and only if Z2 = Z1S−1 for S ∈ Γ(k).

Hence the Kronecker stratum of all codimension

q conditioned invariant subspaces with fixed

restricted Kronecker indices k is diffeomorphic to

M(k, µ)/Γ(k) which is diffeomorphic to an RN.

Hence the Brunovsky strata split in Kronecker cells.

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.17/22

generalised flag manifolds

Given a set of indices 0 ≤ a1 ≤ · · · ≤ as = s the set

Flag(a, Rp) = {(V1, . . . , Vs) ∈

s

• i=1

Gai(Rp) | V1 ⊂ · · · ⊂ Vs}

• f partial flags of type a is a compact analytic

manifold.

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.18/22

generalised flag manifolds

Given a second set of indices 1 ≤ b1 ≤ · · · ≤ bs = p with ai ≤ bi for all i = 1, . . . , s we consider the generalised flag manifold

Flag(a, b, Rp) = {(V1, . . . , Vs) ∈ Flag(a, Rp) | Vi ⊂ span{e1, . . . , ebi}, i = 1, . . . , s}

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.19/22

generalised flag manifolds

Given a second set of indices 1 ≤ b1 ≤ · · · ≤ bs = p with ai ≤ bi for all i = 1, . . . , s we consider the generalised flag manifold

Flag(a, b, Rp) = {(V1, . . . , Vs) ∈ Flag(a, Rp) | Vi ⊂ span{e1, . . . , ebi}, i = 1, . . . , s}

Here the indices ai and bi are the conjugate indices of λi and µi, respectively, read from right to left and aligned properly.

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.19/22

generalised flag manifolds

The manifold Flag(a, b, Rp) is diffeomorphic to a quotient V (a, b)/P(a) where P(a) = V (a, a) is a parabolic group and V (a, b) are the full rank p × p matrices where the last p − bi entries in the columns ai−1 + 1, . . . , ai are zero.

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.20/22

generalised flag manifolds

The manifold Flag(a, b, Rp) is diffeomorphic to a quotient V (a, b)/P(a) where P(a) = V (a, a) is a parabolic group and V (a, b) are the full rank p × p matrices where the last p − bi entries in the columns ai−1 + 1, . . . , ai are zero. There is a cell decomposition of Flag(a, b, Rp) with respect to each reference flag into the subsets with fixed intersection pattern with the reference

• flag. These cell decompositions are known as

Bruhat decompositions.

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.20/22

the retraction map

Theorem [PH]: The surjective smooth and closed maps

γ :M(λ, µ) − → V (a, b) , Z → CZ ¯ C⊤

and

γ :Γ(λ) − → P(a) , S → ¯ CS ¯ C⊤

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.21/22

the retraction map

induce a surjective smooth and closed map

˜ γ : M(λ, µ)/Γ(λ) − → V (a, b)/P(a)

• n quotients, in fact a deformation retract.

A Bruhat type decomposition of the set of conditioned invariant subspaces – p.22/22

the retraction map

induce a surjective smooth and closed map

˜ γ : M(λ, µ)/Γ(λ) − → V (a, b)/P(a)

• n quotients, in fact a deformation retract.

Theorem:

˜ γ maps Kronecker cells onto Bruhat cells with

respect to the reverse standard flag.

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