Cauchy pairs and Cauchy matrices - math.sjtu.edu.cnmath.sjtu.edu.cn/Conference/SCAC/slide/AlisonGordon.pdf · Alison Gordon University of Wisconsin-Madison [email protected] August

Cauchy pairs and Cauchy matrices

Alison Gordon

University of Wisconsin-Madison

[email protected]

August 21, 2012

Alison Gordon (Wisconsin) Cauchy pairs and Cauchy matrices August 21, 2012 1 / 19

Cauchy Pairs

Fix a field K and let V be an K-vector space with finite positive dimension.

Definition

A Cauchy pair on V is an ordered pair X ,Y ∈ End(V ) which satisfy thefollowing conditions:

(i) Both X and Y are diagonalizable.

(ii) X − Y is rank 1.

(iii) There does not exist a subspace W of V such thatXW ⊆W ,YW ⊆W ,W 6= 0,W 6= V .

We call these pairs Cauchy due to the way they arise from Cauchymatrices.


Cauchy Matrices

Definition

A Cauchy matrix is a matrix C ∈ Matn(K) with (i,j)-entry:

Cij =1

xi − yj

for 1 ≤ i , j ≤ n, where x1, . . . , xn, y1, . . . , yn ∈ K are pairwise distinct.

Example: Taking x1 = 1, x2,= 3, x3 = 5, y1 = 0, y2 = 2, y3 = 4,1 −1 −13

13 1 −115

13 1

=

11−0

11−2

11−4

13−0

13−2

13−4

15−0

15−2

15−4


Cauchy Determinant Identity

Theorem (Cauchy, 1841)

Let C ∈ Matn(K) be Cauchy with Cij = 1xi−yj

. Then

det C =

∏ni=2

∏i−1j=1(xi − xj)(yj − yi )∏n

i=1

∏nj=1(xi − yj)

Note that det C 6= 0, so C is invertible.


Displacement Equation

For the Cauchy matrix C , we define matrices X and Y as:

X =

x1 0 · · · 00 x2 · · · 0...

.... . . 0

0 0 · · · xn

Y =

y1 0 · · · 00 y2 · · · 0...

.... . . 0

0 0 · · · yn

C satisfies the displacement equation

XC − CY = J

where J is the matrix of all ones.


Displacement Equation

For the Cauchy matrix C , we define matrices X and Y as:

X =

x1 0 · · · 00 x2 · · · 0...

.... . . 0

0 0 · · · xn

Y =

y1 0 · · · 00 y2 · · · 0...

.... . . 0

0 0 · · · yn

C satisfies the displacement equation

XC − CY = J

where J is the matrix of all ones.


Constructing Cauchy Pairs

Setup:

Let V be a K-vector space with dimension n ≥ 2.

Let C be a Cauchy matrix with (i,j)-entry 1xi−yj

for 1 ≤ i , j ≤ n.

Fix a bases BX ,BY for V such that C is the transition matrix fromBX to BY .

Define matrices X , Y as in the previous slide.

Define X ∈ End(V ) such that X acts as X with respect to BX .

Define Y ∈ End(V ) such that Y acts as Y with respect to BY .



Setup:

Let V be a K-vector space with dimension n ≥ 2.

Let C be a Cauchy matrix with (i,j)-entry 1xi−yj

for 1 ≤ i , j ≤ n.

Fix a bases BX ,BY for V such that C is the transition matrix fromBX to BY .

Define matrices X , Y as in the previous slide.

Define X ∈ End(V ) such that X acts as X with respect to BX .

Define Y ∈ End(V ) such that Y acts as Y with respect to BY .



By construction, X and Y are both diagonalizable.

With respect to BY , X acts as

C−1XC = Y + C−1J

by the displacement equation.

Thus, (X − Y ) acts as C−1J with respect to BY , so (X − Y ) is rank 1.



By construction, X and Y are both diagonalizable.

With respect to BY , X acts as

C−1XC = Y + C−1J

by the displacement equation.

Thus, (X − Y ) acts as C−1J with respect to BY , so (X − Y ) is rank 1.



Any X -invariant and Y -invariant subspace of V must be:

The span of some subset of BY and

Invariant under (X − Y ), represented by C−1J with respect to BY .

Since entries of C−1J are non-zero, (X − Y )V is not contained in thespan of any proper subset of BY . Thus, no such proper subspace exists.

Therefore (X ,Y ) is a Cauchy pair.



Any X -invariant and Y -invariant subspace of V must be:

The span of some subset of BY and

Invariant under (X − Y ), represented by C−1J with respect to BY .

Since entries of C−1J are non-zero, (X − Y )V is not contained in thespan of any proper subset of BY . Thus, no such proper subspace exists.

Therefore (X ,Y ) is a Cauchy pair.


Cauchy pairs from Cauchy matrices

Proposition

Let V be an n-dimensional K-vector space and let C ∈ Matn(K) be aCauchy matrix. Then there exists a Cauchy Pair (X,Y) with eigenbasesBX ,BY such that C is the transition matrix from BX to BY .

This proposition follows from the previous construction.


Next Goal

The previous construction shows how a Cauchy pair arises from a Cauchymatrix.

This map, sending a Cauchy matrix to a Cauchy pair, produces a naturalcorrespondence between Cauchy matrices and Cauchy pairs.

To show this, we must find an inverse map that shows how Cauchymatrices arise from Cauchy pairs.


Next Goal

Goal

Show that, for every Cauchy pair (X ,Y ), there exists a pair of eigenbasesfor X ,Y such that the transition matrix between them is Cauchy.

First, we need to establish a few facts about Cauchy pairs.


Eigenvalues of Cauchy pairs

Lemma

Let (X ,Y ) be a Cauchy pair on V . Then X and Y have no commoneigenvalues.

To prove this, it suffices to show that for an eigenbasis {vi}ni=1 for X ,

W = span{vi |(λI − X )vi 6= 0}

is X - and Y -invariant, whenever λ is a eigenvalue for Y .

Thus, if λ is an eigenvalue for Y , it cannot be an eigenvalue for X .



Lemma

Let (X ,Y ) be a Cauchy pair on V . Then X and Y have no commoneigenvalues.

To prove this, it suffices to show that for an eigenbasis {vi}ni=1 for X ,

W = span{vi |(λI − X )vi 6= 0}

is X - and Y -invariant, whenever λ is a eigenvalue for Y .

Thus, if λ is an eigenvalue for Y , it cannot be an eigenvalue for X .



Lemma

Let (X ,Y ) be a Cauchy pair on V . Then X and Y are bothmultiplicity-free.

These two lemmas show that, for a Cauchy pair (X ,Y ), the eigenvalues ofX and the eigenvalues of Y form a set of 2n pairwise distinct scalars.



Lemma

Let (X ,Y ) be a Cauchy pair on V . Then X and Y are bothmultiplicity-free.

These two lemmas show that, for a Cauchy pair (X ,Y ), the eigenvalues ofX and the eigenvalues of Y form a set of 2n pairwise distinct scalars.


Notation

Fix a Cauchy pair (X ,Y ) on V .

Let {xi}ni=1 and {yi}ni=1 be orderings of the eigenvalues of X and Y ,respectively.

Fix a basis of eigenvectors {vi}ni=1 for X such thatn∑i=j

vj ∈ (X − Y )V .

We call such a basis a normalized eigenbasis for X .

Then there exist scalars {αi}ni=1 such that (X − Y )vi = αi

n∑j=1

vj .


Notation

Fix a Cauchy pair (X ,Y ) on V .

Let {xi}ni=1 and {yi}ni=1 be orderings of the eigenvalues of X and Y ,respectively.

Fix a basis of eigenvectors {vi}ni=1 for X such thatn∑i=j

vj ∈ (X − Y )V .

We call such a basis a normalized eigenbasis for X .

Then there exist scalars {αi}ni=1 such that (X − Y )vi = αi

n∑j=1

vj .


The scalars αi

The scalars {αi}ni=1 have the following interesting connection with theeigenvalues of X ,Y :

Lemma

For 1 ≤ j ≤ n,n∑

i=1

αi

xi − yj= 1

For 1 ≤ j ≤ n, we define

wj =n∑

i=1

1

xi − yjvi


The scalars αi

The scalars {αi}ni=1 have the following interesting connection with theeigenvalues of X ,Y :

Lemma

For 1 ≤ j ≤ n,n∑

i=1

αi

xi − yj= 1

For 1 ≤ j ≤ n, we define

wj =n∑

i=1

1

xi − yjvi


An eigenbasis for Y

Proposition

For 1 ≤ i ≤ n, the vector wi is an eigenvector for Y with eigenvalue yi .Thus, {wi}ni=1 is an eigenbasis for Y .

Corollary

The transition matrix from {vi}ni=1 to {wi}ni=1 is Cauchy with (i , j)-entry1

xi−yj


Cauchy transition matrices

Theorem

Let V be an n-dimensional K-vector space and let (X ,Y ) be a CauchyPair on V . Then there exist an eigenbasis BX for X and an eigenbasisBY for Y such that the transition matrix from BX to BY is Cauchy.


A bijection between Cauchy pairs and Cauchy matrices

In conclusion, we have shown that:

Given a Cauchy matrix C , there exists a Cauchy pair (X ,Y ) for whichC is the transition matrix between eigenbases for X and Y .

Given a Cauchy pair (X ,Y ), there exists a Cauchy matrix C which isa transition matrix between eigenbases for X and Y .

Together, this gives a bijection, up to normalization, between the class ofCauchy matrices and the class of Cauchy pairs.


A bijection between Cauchy pairs and Cauchy matrices

In conclusion, we have shown that:

Given a Cauchy matrix C , there exists a Cauchy pair (X ,Y ) for whichC is the transition matrix between eigenbases for X and Y .

Given a Cauchy pair (X ,Y ), there exists a Cauchy matrix C which isa transition matrix between eigenbases for X and Y .

Together, this gives a bijection, up to normalization, between the class ofCauchy matrices and the class of Cauchy pairs.


Thank You!


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Cauchy pairs and Cauchy matrices - math.sjtu.edu.cnmath.sjtu.edu.cn/Conference/SCAC/slide/AlisonGordon.pdf · Alison Gordon University of Wisconsin-Madison [email protected] August