A PARAMETRIZATION OF MATRIX CONJUGACY ORBIT SETS AS UNIONS OF AFFINE PLANES

PETERIS DAUGULIS^{∗}

Abstract. The problem of finding a canonical form of complex matrices up to conjugacy with the set of canonical matrices being a union of affine planes in the matrix space is considered. A solution of the problem is given producing a new canonical form for matrices over algebraically closed fields.

Key words. Jordan form, rational form, eigenvalue, Weyr characteristic, affine plane.

MSC 15A21.

1. Introduction. It is known that sets of representatives of matrix conjugacy classes in Jordan or rational canonical (elementary divisors version) forms in general do not constitute algebraic varieties in the ambient matrix spaces.

Example 1.1. Consider 2 × 2 matrices over ℂ. A set of the Jordan forms can be chosen as J_{1} ∪ J_{2} where

J_{1} = \{[ \begin{matrix}a \\ & b \end{matrix} ] , ∀ a, b ∈ ℂ\}, J_{2} = \{\left[{c \atop 1 c}\right] , ∀ c ∈ ℂ\}.

If a ≠ b then matrices [ \begin{matrix}a \\ & b \end{matrix} ] and [ \begin{matrix}b \\ & a \end{matrix} ] are conjugate therefore J_{1} contains

two representatives of each matrix conjugacy class with two distinct eigenvalues.

M.L.Kontsevich [1], p.127, has proposed the problem of finding a canonical form of matrices over ℂ for which the set of representatives of matrix conjugacy classes would be a union of disjoint affine planes - cosets in the ambient ℂ-linear matrix space modulo a subspace. In other words, for each n ≥ 2 find a union of disjoint affine planes A in the ℂ-linear space Mat(n, n, ℂ) ≃ ℂ^{n^{2}} so that each matrix conjugacy orbit intersects A in exactly one point. The problem may be motivated by a desire to provide more links between the representation theory of algebras and algebraic geometry.

Note that neither the Jordan form nor the rational canonical forms satisfy this

∗Department of Mathematics, Daugavpils University, Daugavpils, Parades 1, Latvia (peteris.daugulis@du.lv).

condition. The rational canonical form with invariant factors may be investigated if one wants to use affine varieties of higher degrees but this is not within the scope of this paper.

In this paper we present a solution of the above problem by developing a new canonical form of matrices which we call canonical plane form. Matrices considered are defined over ℂ. Similar results can be proved for any algebraically closed field.

2. Description of canonical plane matrices and affine planes. The goal of this section is to describe complex matrices called canonical plane matrices which will be interpreted as points of affine planes in the ℂ-linear matrix spaces. In the first step we define matrices over multivariate polynomial rings. Then we obtain complex matrices using substitutions.

Matrices over polynomial rings are denoted using bold letters. E_{m} denotes the identity matrix.

2.1. Matrices over multivariate polynomial rings.

2.1.1. Partitions. Given n ∈ ℕ, n ≥ 2, we consider all (nonincreasing) partitions of n, i.e. sequences of natural numbers - addends (n_{1}, ..., n_{d}) such that n_{i} ≥ n_{i+1},

∀ i : 1 ≤ i ≤ d − 1 and ∑_{i=1}^{d} n_{i} = n. Maximal constant contiguous subsequences, i.e.

maximal subsequences (n_{r}, ..., n_{r+s}) with n_{r} = ... = n_{r+s} will be called stacks. A partition of n having t stacks of lengths l_{1}, ..., l_{t} with distinct addends m_{1} > m_{2} > ... > m_{t} may be denoted, symbolically, as (m_{1}^{l_{1}}, ..., m_{t}^{l_{t}}).

2.1.2. Diagonal matrices. Given a partition π = (n_{1}, n_{2}, ..., n_{d}) of n define a diagonal matrix

D_{π} = ⨁_{i=1}^{d} X_{i} · E_{n_{i}} ∈ Mat(n, n, ℂ[X_{1}, ..., X_{d}]).

We use the convention that indices of the variables in diagonal blocks are arranged in the increasing order starting from the upper left corner as shown in the example.

A note on generalization of eigenvector centrality for bipartite graphs and applications 3

Example 2.1. Let n = 8, π = (3, 3, 2), then

D_{π} = X_{1}E_{3} ⊕ X_{2}E_{3} ⊕ X_{3}E_{2} = [ \begin{matrix}X_{1} \\ & X_{1} \\ & & X_{1} \\ & & & X_{2} \\ & & & & X_{2} \\ & & & & & X_{2} \\ & & & & & & X_{3} \\ & & & & & & & X_{3} \end{matrix} ] .

2.1.3. Generalized companion matrices. For any l ≤ d define a l × l matrix R_{l}(X_{i_{1}}, ..., X_{i_{l}}) as follows:

R_{l}(X_{i_{1}}, ..., X_{i_{l}}) = [ \begin{matrix}0 & 0 & ... & 0 & (−1)^{l+1}e_{l}(X_{i_{1}}, ..., X_{i_{l}}) \\ 1 & 0 & ... & 0 & (−1)^{l}e_{l−1}(X_{i_{1}}, ..., X_{i_{l}}) \\ 0 & 1 & ... & 0 \\ ... & ... & ... & ... & ... \\ 0 & 0 & ... & 1 & e_{1}(X_{i_{1}}, ..., X_{i_{l}}) \end{matrix} ]

where e_{i} is the elementary symmetric polynomial of degree i in l variables. It is the l × l companion matrix having l distinct ”eigenvalues” X_{i_{1}}, ..., X_{i_{l}}.

Example 2.2. R_{3}(X_{1}, X_{2}, X_{3}) = [ \begin{matrix}0 & 0 & X_{1}X_{2}X_{3} \\ 1 & 0 & −X_{1}X_{2} − X_{1}X_{3} − X_{2}X_{3} \\ 0 & 1 & X_{1} + X_{2} + X_{3} \end{matrix} ].

2.1.4. Block lower triangular matrices. Given a partition of π = (n_{1}, ..., n_{d}) of n define a lower block triangular matrix P_{π} by modifying D_{π} in two steps as follows:

1. for each pair (n_{j}, n_{j+1}) with n_{j} > n_{j+1} insert the matrix
F_{n_{j+1},n_{j}} = [O_{n_{j+1},n_{j}−n_{j+1}}|E_{n_{j+1}}] as shown:

[ \frac{X_{j} · E_{n_{j}}}{X_{j+1} · E_{n_{j+1}}} ] ⟶ [ \frac{X_{j} · E_{n_{j}}}{F_{n_{j+1},n_{j}} X_{j+1} · E_{n_{j+1}}} ] ,

2. for each stack (n_{r}, n_{r+1}, ..., n_{r+u}) of π with n_{r} = ... = n_{r} = m substitute the submatrix

[ \begin{matrix}X_{r} · E_{m} \\ & X_{r+1} · E_{m} \\ & & X_{r+2} · E_{m} \\ & & & ... \\ & & & & X_{r+u} · E_{m} \end{matrix} ]

by

R_{u+1}(X_{r}, ..., X_{r+u}) ⊗ E_{m} =

[ \begin{matrix}& & & (−1)^{u+2}e_{u+1}(X_{r}, ..., X_{r+u}) · E_{m} \\ E_{m} & & & (−1)^{u+1}e_{u}(X_{r}, ..., X_{r+u}) · E_{m} \\ E & m \\ & & ... \\ & & E_{m} & e_{1}(X_{r}, ..., X_{r+u}) · E_{m} \end{matrix} ]

Thus stacks of π bijectively correspond to diagonal blocks of P_{π} and off-diagonal blocks of P_{π} are identity matrices coming from F_{n_{j+1},n_{j}}.

Example 2.3. Let n = 8, π = (3, 3, 2), then

P_{π} = [ \begin{matrix}& & & −X_{1}X_{2} \\ & & & & −X_{1}X_{2} \\ & & & & & −X_{1}X_{2} \\ 1 & & & X_{1} + X_{2} \\ & 1 & & & X_{1} + X_{2} \\ & & 1 & & & X_{1} + X_{2} \\ & & & & 1 & & X_{3} \\ & & & & & 1 & & X_{3} \end{matrix} ] .

2.2. Complex canonical plane matrices and affine planes.

2.2.1. Substitutions maps. Define the substitution map

φ : Mat(n, n, ℂ[X_{1}, ..., X_{d}]) × ℂ^{d} → Mat(n, n, ℂ),

([f_{ij}], (a_{1}, ..., a_{d})) ↦ [f_{ij}(a_{1}, ..., a_{d})].

Denote the image of P_{π} × ℂ^{d} under φ by P_{π}.

2.2.2. Description of canonical plane matrices. Let f = X^{l} − ∑_{i=0}^{l−1} a_{i}X^{i} ∈

ℂ[X]. Define a companion matrix C_{f} of f in a standard form:

C_{f} = C(a_{0}, ..., a_{l−1}) = [ \begin{matrix}0 & 0 & ... & 0 & a_{0} \\ 1 & 0 & ... & 0 & a_{1} \\ 0 & 1 & ... \\ ... & ... & ... & ... & ... \\ 0 & 0 & ... & 1 & a_{l−1} \end{matrix} ]

Each A ∈ P_{π} is a block lower triangular matrix with the diagonal blocks in form C(a_{0}, ..., a_{l−1}) ⊗ E_{m} for some l and (a_{0}, ..., a_{l−1}) ∈ ℂ^{l}.

Example 2.4. Let n = 8, π = (3, 3, 2), then

A = [ \begin{matrix}& & & −2 \\ & & & & −2 \\ & & & & & −2 \\ 1 & & & 3 \\ & 1 & & & 3 \\ & & 1 & & & 3 \\ & & & & 1 & & 1 \\ & & & & & 1 & & 1 \end{matrix} ] = φ(P_{π}, (1, 2, 1)).

2.2.3. Inverse image of a canonical plane matrix. Given A ∈ P_{π} we can find an element of its inverse image under φ as follows. For each diagonal block

C(a_{0}, ..., a_{l−1}) ⊗ E_{m} of A solve the equation X^{l} − ∑_{i=0}^{l−1} a_{i}X^{i} = 0, get the multiset of

roots [λ_{1}, ..., λ_{l}], its elements are the complex numbers which must be substituted (in any order) into the polynomial arguments in the corresponding diagonal block of P_{π}. Note that the vector (a_{0}, ..., a_{l−1}) uniquely determines the multiset of roots [λ_{1}, ..., λ_{l}] and vice versa.

2.2.4. Canonical affine planes. Affine planes in a linear space L are identified with additive cosets modulo a subspace V of L. Thus to define an affine plane P ⊆ L we need to fix one element l ∈ P (a constant shift) and describe V (linear part). Dimension of P is equal to dim V.

Theorem 2.5. n ∈ ℕ, n ≥ 2, π = (n_{1}, ..., n_{d}) - a partition of n. Then P_{π} is an affine plane in Mat(n, n, ℂ) of dimension d.

Proof. We will express A ∈ P_{π} as a sum

A = S + L(A)

of a constant matrix S (the shift) and a variable matrix L(A) (the linear part) as follows. The nonzero elements of S are the off-diagonal blocks together with the identity submatrices of generalized companion matrices. Thus S is uniquely defined for all matrices in P_{π}. We define L(A) = A−S and observe that the possibly nonzero elements of L(A) are the blocks of form [a_{0}|...|a_{l−1}]^{T} ⊗ E_{m} corresponding to last columns of generalized companion matrices.

Supppose A ∈ P_{π} has the t diagonal blocks C(a_{j0}, ..., a_{j,l_{j}−1}) ⊗ E_{m_{j}}, ∀j ∈

{1, ..., t}. For each j ∈ {1, ..., t} solve the j-th stack equation X^{l_{j}} − ∑_{i=0}^{l_{j}−1} a_{ji}X^{i} = 0,

get the multiset of roots [λ_{j1}, ..., λ_{j,l_{j}}], ∀j ∈ {1, ..., t}. We have that

A = φ_{M}(P_{π}, (λ_{11}, ..., λ_{1,l_{1}}, ..., λ_{t1}, ..., λ_{t,l_{t}})).

Note that the ordering of the roots within each stack does not change the value of φ. We see that A ∈ P_{π} can be constructed for any sequence of vectors (a_{10}, ..., a_{1,l_{1}−1}), ..., (a_{t0}, ..., a_{t,l_{t}−1}) thus A − S runs through a linear subspace V_{π} of Mat(n, n, ℂ) as A runs through P_{π}. We see that dim V_{π} is the sum of the dimensions of vectors (a_{10}, ..., a_{1,l_{1}−1}),...,(a_{t0}, ..., a_{t,l_{t}−1}) which is equal to d.

Example 2.6. Consider the matrix A of example 2.4. In this case

S = [ \begin{matrix}1 \\ & 1 \\ & & 1 \\ & & & 1 \\ & & & & 1 \end{matrix}  ] , L(A) = [  \begin{matrix}−2 \\ & −2 \\ & & −2 \\ 3 \\ & 3 \\ & & 3 \\ & & & 1 \\ & & & & 1 \end{matrix} ] .

dim V_{π} = 3, V_{π} = ⟨V_{1}, V_{2}, V_{3}⟩ where

V_{1} = [  \begin{matrix}1 \\ & 1 \\ & & 1 \end{matrix}  ] , V_{2} = [  \begin{matrix}1 \\ & 1 \\ & & 1 \end{matrix}  ] ,

V_{3} = [ \begin{matrix}1 \\ & 1 \end{matrix} ] .

Theorem 2.7. n ∈ ℕ, n ≥ 2, π and π′ - distinct partitions of n. Then

P_{π} ∩ P_{π′} = ∅.

Proof. We show that supports of constant shift matrices are different. Given π ≠ π′ with π = (n_{1}, n_{2}, ...) and π′ = (n′_{1}, n′_{2}, ...) consider the distinct pair n_{j} ≠ n′_{j} with the minimal j. There are two possibilities: (1) new stacks start from n_{j} and n′_{j} or (2) a new stack starts from one of n_{j} or n′_{j}. In the first case the off-diagonal blocks of P_{π} and P_{π′} inserted to the left of diagonal blocks for the new stacks have 1’s in different positions. In the second case in P_{π} and P_{π′} we have diagonal blocks of distinct sizes having the same position of upper left corners, by considering identity matrix blocks of generalized companion matrices it follows that at least one 1’ in the off-diagonal block below the shorter diagonal block in, say, P_{π} is absent in P_{π′}.

3. Main results.

3.1. Jordan forms of canonical plane matrices.

3.1.1. Preparations. Let a partition π have t stacks of lengths l_{1}, ..., l_{t} with

distinct addends m_{1} > m_{2} > ... > m_{t}. Define m_{t+1} = 0. Define s_{j} = ∑_{i=1}^{j} l_{i}. Define

P_{j} = {m_{1} − m_{j} + 1, ..., m_{1} − m_{j+1}}.

Let A ∈ P_{π} have diagonal blocks C(a_{j0}, ...a_{j,l_{j}−1}) ⊗ E_{m_{j}}, j ∈ {1, ..., t}. For each j ∈ {1, ..., t} solve the jth stack equation

X^{l_{j}} − ∑_{i=0}^{l_{j}−1} a_{ji}X^{i} = 0,

get the multiset of roots [λ_{j1}, ..., λ_{j,l_{j}}],j ∈ {1, ..., t}. Denote the muliplicity of λ as a

root for the ith stack equation by μ(λ, i). Denote ∑_{i=1}^{j} μ(λ, i) by α(λ, j).

We think of A acting in ℂ^{n} - the n × 1 column space with the standart basis B = {e_{1}, ..., e_{n}}, e_{i} = E_{i1} where E_{ij} is a matrix unit.

The goal of this subsection is to find the Jordan form of A ∈ P_{π}. This is done in several steps:

1. by inspecting the digraph of A we find a decomposition of ℂ^{n} into a direct sum of m_{1} A-invariant subspaces which are generated by e_{1},...,e_{m_{1}} as ℂ[A]- modules,

2. decompose A into a direct sum by permutations,

3. decompose the obtained direct summands of A into the Jordan form using their block structure.

3.1.2. Invariant subspaces. For each i ∈ {1, ..., m_{1}} define a A-invariant subspace V_{i} = ℂ[A] · e_{i}. For each j ∈ {1, ..., t} define a s_{j} × s_{j} matrix

G_{j} = ⨁_{i=1}^{j} C(c_{i0}, ..., c_{i,l_{i}−1}) + ∑_{i=1}^{j−1} E_{s_{i}+1,s_{i}}.

Here we also use the convention about the block ordering as in 2.1.2, see example 3.2 below.

Theorem 3.1. In the notations of 3.1.1 we have

1. ℂ^{n} = ⨁_{i=1}^{m_{1}} V_{i},

2. dim V_{i} = s_{j} for i ∈ P_{j},

3. If i ∈ P_{j} then the restriction of A to V_{i} with respect to B is G_{j}.

Proof. The statements are proved by considering the images of e_{1}, ..., e_{m_{1}} under powers of A, using the digraph of A and induction. The digraph of A with respect to B decomposes into m_{1} weakly connected components corresponding to V_{1}, ..., V_{m_{1}}. Vertex sets of these components and restrictions of A can be explicitly described to prove the restriction statement. The number of elements of B in the component corresponding to V_{i} is equal to s_{j} if i ∈ P_{j} which implies the dimension statement. Further details are omitted.

Example 3.2. Consider the matrix A of example 2.4. Its digraph is shown in Fig.1.

[figure: e_{1}; e_{4}; 3; −2; e_{2}; e_{5}; 3; −2; e_{7}; e_{3}; e_{6}; 3; −2; e_{8}]

Fig.1. - the digraph for A of example 2.4

In this case ℂ^{8} = V_{1} ⊕ V_{2} ⊕ V_{3} where V_{1} = ⟨e_{1}, e_{4}⟩, V_{2} = ⟨e_{2}, e_{5}, e_{7}⟩, V_{3} = ⟨e_{3}, e_{6}, e_{8}⟩.

G_{1} = [ \begin{matrix}0 & −2 \\ 1 & 3 \end{matrix} ], G_{2} = [ \begin{matrix}0 & −2 & 0 \\ 1 & 3 & 0 \\ 0 & 1 & 1 \end{matrix} ].

3.1.3. Decomposition by permutation. Denote the matrix conjugacy relation by ≃.

Theorem 3.3. In the notations of 3.1.1 we have

A ≃ ⨁_{j=1}^{t}(m_{j} − m_{j+1})G_{j}.

Proof. The described direct sum is obtained by permuting the rows and columns of A following theorem 3.1.

Example 3.4. The matrix A of example 2.4 is permutation conjugate to

G_{1} ⊕ G_{2} ⊕ G_{2} = [ \begin{matrix}& −2 \\ 1 & 3 \\ & & & −2 \\ & & 1 & 3 \\ & & & 1 & 1 \\ & & & & & & −2 \\ & & & & & 1 & 3 \\ & & & & & & 1 & 1 \end{matrix} ] .

3.1.4. Jordan forms. Denote the Jordan block with eigenvalue λ and size i by J_{i}(λ).

Theorem 3.5. In the notations of 3.1.1 we have

G_{j} ≃ ⨁_{λ} J_{α(λ,j)}(λ).

Proof. The result follows from the lower triangular block structure of G_{j}. G_{j} is in block lower triangular form therefore its characteristic polynomial χ(G_{j}, x) is equal to the product of the characteristic polynomials of diagonal blocks. It follows that χ(G_{j}, x) = ∏_{λ}(λ − x)^{α(λ,j)}. The diagonal blocks of G_{j} have subdiagonals with all

elements equal to 1 and only zeros under subdiagonals. The only nonzero elements outside diagonal blocks of G_{j} are 1’s in the corner positions between diagonal blocks. It follows that G_{j} has the global nonzero subdiagonal. It follows that G_{j} has a cyclic vector [1|0|0|...|0]^{T}. Hence the minimal polynomial of G_{j} is ±χ(G_{j}, x). It follows G_{j} has one Jordan block for each eigenvalue.

Theorem 3.6. In the notations of 3.1.1 we have

A ≃ ⨁_{j=1}^{t}(m_{j} − m_{j+1})( ⨁_{λ} J_{α(λ,j))}(λ)). (3.1)

Proof. It follows from theorems 3.3 and 3.5.

Example 3.7. The Jordan form of A from example 2.4 is shown in Fig.2.

[figure: e′_{1}; 2; e′_{4}; e′_{2}; 2; e′_{5}; e′_{7}; e′_{3}; 2; e′_{6}; e′_{8}]

Fig.2. - the Jordan form digraph of A from example 2.4

3.2. Canonical plane forms of Jordan matrices. The goal of this subsection is to show that a direct sum of Jordan matrices is conjugate to a canonical plane matrix.

3.2.1. Weyr characteristic of canonical plane matrices. The Jordan form of a matrix B with the single eigenvalue λ is determined by its Weyr characteristic sequence Ω(B, λ) = (ω_{1}, ω_{2}, ...) where ω_{i} is the number of Jordan blocks of size at least i (see [2] for a discussion and recent applications).

Example 3.8. If B ≃ 2J_{1}(λ) ⊕ J_{3}(λ) ⊕ J_{4}(λ) then Ω(B, λ) = (4, 2, 2, 1, 0, ...), ω_{u} = 0 for all u ≥ 5. The Jordan form of any matrix B with eigenvalues λ_{1}, λ_{2}, ... is determined by the Weyr array Ω(B) = (Ω(B, λ_{1}), Ω(B, λ_{2}), ...).

Theorem 3.9. In the notations of 3.1.1 let an eigenvalue λ of A have nonzero multiplicity μ(λ, j) if and only if j ∈ {j_{1}, j_{2}, ..., j_{v}}, j_{1} < j_{2} < .. < j_{v} Then

Ω(A, λ) = (\underbrace{m_{j_{1}}, ..., m_{j_{1}}}_{μ(λ,j_{1}) copies}, \underbrace{m_{j_{2}}, ..., m_{j_{2}}}_{μ(λ,j_{2}) copies}, ..., \underbrace{m_{j_{v}}, ..., m_{j_{v}}}_{μ(λ,j_{v}) copies}, 0, ...).

Proof. The statement follows from theorem 3.6 by reordering the direct sum (3.1), counting the number of Jordan blocks of A of given λ and size and using induction. We have that

A ≃ ⨁_{j=1}^{t}(m_{j} − m_{j+1})( ⨁_{λ} J_{α(λ,j)}(λ)) ≃

⨁_{λ} ( ⨁_{j=1}^{t}(m_{j} − m_{j+1})J_{α(λ,j)}(λ)) =

⨁_{λ} ( ⨁_{j∈{j_{1},j_{2},...,j_{v}}}(m_{j} − m_{j+1})J_{α(λ,j)}(λ)).

Thus for a given λ

1. the minimal size of a Jordan block is α(λ, j_{1}) = μ(λ, j_{1}), there are m_{j_{1}} Jordan

blocks of sizes at least 1, ..., μ(λ, j_{1}), start Ω(A, λ) as \underbrace{(m_{j_{1}}, ..., m_{j_{1}})}_{μ(λ,j_{1}) copies},

2. the next possible size of a Jordan block is α(λ, j_{2}), there are m_{j_{1}} − (m_{j_{1}} − m_{j_{2}}) = m_{j_{2}} Jordan blocks of sizes at least μ(λ, j_{1}) + 1, ..., α(λ, j_{2}), continue

Ω(A, λ) as (\underbrace{m_{j_{1}}, ..., m_{j_{1}}}_{μ(λ,j_{1}) copies}, \underbrace{m_{j_{2}}, ..., m_{j_{2}}}_{μ(λ,j_{2}) copies}),

3. ...

3.2.2. From Jordan forms to canonical plane matrices. We describe an algorithm for finding a canonical plane matrix which is conjugate to a given matrix B. Rearrange Ω(B) array as follows:

1. sort the nonzero entries of Ω(B) in nonincreasing order,

2. collect the equal nonzero entries into a sequence of multisets

M(B) = (M_{1}, M_{2}, ..., M_{t}),

all elements of M_{i} are equal to some m_{i}, m_{1} > m_{2} > ... > m_{t}.

Denote |M_{i}| by l_{i}, denote the multisets of eigenvalues corresponding to the elements of M_{i} by Λ_{i} = [λ_{i1}, ..., λ_{i,l_{i}}]. Define π = (m_{1}^{l_{1}}, ..., m_{t}^{l_{t}}).

Theorem 3.10. In the above notations

B ≃ B̃ = φ(P_{π}, (\underbrace{λ_{11}, ..., λ_{1,l_{1}}}_{1st block}, ..., \underbrace{λ_{t1}, ..., λ_{t,l_{t}}}_{t−th block})).

Proof. Using the theorem 3.9 we construct Ω(B̃) by considering diagonal blocks of B̃:

1. from the 1st block we get the initial elements equal to m_{1} of Ω(B̃, λ) for λ ∈ Λ_{1}, thus get initial subsequences of Ω(B̃, λ) of form \underbrace{(m_{1}, ..., m_{1})}_{μ(λ,1)} for each

λ with μ(λ, 1) ≠ 0,

2. from the 2nd block we get elements equal to m_{2} of Ω(B̃, λ) for λ ∈ Λ_{2}, thus form subsequences of Ω(B̃, λ) of length α(λ, 2) by adding \underbrace{(m_{2}, ..., m_{2})}_{μ(λ,2) times} for each

λ with μ(λ, 1) ≠ 0, μ(λ, 2) ≠ 0 or initiate Ω(B̃, λ) as \underbrace{(m_{2}, ..., m_{2})}_{μ(λ,2) times} for each λ

with μ(λ, 1) = 0, μ(λ, 2) ≠ 0.

3. ...

By induction on |M| we see that Ω(B̃) = Ω(B) therefore B̃ ≃ B.

Definition 3.11. For any matrix B denote its canonical plane matrix B̃ by Aff(B).

Remark 3.12. Note that the canonical plane form of a matrix in general is different from the Jordan form and the rational form (either in invariant factors or elementary divisors version) as example 2.4 shows.

3.3. Bijectivity of the correspondence between canonical plane matrices and Jordan forms.

Theorem 3.13.

1. A, A′ - canonical plane n × n matrices. Then

A ≠ A′ ⟹ A ≄ A′.

2. B,B′ - any n × n matrices. Then

B ≄ B′ ⟹ Aff(B) ≠ Aff(B′).

Proof.

1. Let A ∈ P_{π}, A′ ∈ P_{π′}, A ≠ A′. If π ≠ π′ then the sets of addends of π and π′ are different which by theorem 3.9 implies Ω(A) ≠ Ω(A′). If π = π′ then A = S + L(A), A′ = S + L(A′) as in the proof of theorem 2.5. It follows that L(A) ≠ L(A)′ hence for at least one λ ∈ ℂ the multiplicity functions μ(λ, i) for A and A′ are different. By theorem 3.9 it folows that Ω(A, λ) ≠ Ω(A′, λ).

2. Let B ≄ B′. Then Ω(B) ≠ Ω(B′). Hence there exist m and λ ∈ ℂ such that the multiplicities of m in Ω(B, λ) and Ω(B′, λ) are different. Considering M(B) and M(B′) as explained at the beginning of 3.2.2 and theorem 3.10 we see that L(B) ≠ L(B′). It follows that Aff(B) ≠ Aff(B′).

3.4. Conclusion. For any n ∈ ℕ, n ≥ 2, define P = ⋃_{π} P_{π}, where the union is

taken over all partitions of n.

Theorem 3.14. P contains each n × n matrix conjugacy class exactly once.

Proof. The statement follows from theorems 3.10 and 3.13.

3.5. Appendix - canonical plane matrices for two values of n.

3.5.1. n = 2. There are two partitions of 2: π_{1} = (2) and π_{2} = (1, 1). The affine planes are

1. P_{π_{1}} = \{[ \frac{a}{a} ] , ∀ a ∈ ℂ\}, each matrix in P_{π_{1}} is in its Jordan form 2J_{1}(a),

2. P_{π_{2}} = \{[ \frac{a}{1 b} ] , ∀ a, b ∈ ℂ\}, the Jordan form depends on the roots of the

characteristic polynomial x^{2} − bx − a:

(a) if there are 2 simple roots λ_{1}, λ_{2} then J_{1}(λ_{1}) ⊕ J_{1}(λ_{2}),

(b) if there is 1 double root λ_{0} then J_{2}(λ_{0}).

3.5.2. n = 5. We describe all canonical plane matrices for n = 5 and give the corresponding Jordan forms. One can check that there are 27 subcases which correspond to the 27 distinct Jordan forms for 5 × 5 complex matrices. Below for each subcase distinct arguments for polynomial roots and Jordan blocks mean distinct complex numbers.

There are seven partitions of 5: π_{1} = (5), π_{2} = (4, 1), π_{3} = (3, 2), π_{4} = (3, 1, 1), π_{5} = (2, 2, 1), π_{6} = (2, 1, 1, 1), π_{7} = (1, 1, 1, 1, 1). For each partition π we give the description of P_{π}:

1. P_{π_{1}} = \{[ \begin{matrix}a \\ & a \\ & & a \\ & & & a \\ & & & & a \end{matrix} ] , ∀ a ∈ ℂ\}, each matrix in P_{π_{1}} is in its Jordan

form 5J_{1}(a);

2. P_{π_{2}} = \{[ \begin{matrix}a \\ & a \\ & & a \\ & & & a \\ & & & 1 & b \end{matrix} ] , ∀ a, b ∈ ℂ\}, the Jordan forms:

(a) 4J_{1}(a) ⊕ J_{1}(b) if a ≠ b,

(b) 3J_{1}(a) ⊕ J_{2}(a) if a = b;

3. P_{π_{3}} = \{[ \begin{matrix}a \\ & a \\ & & a \\ & 1 & & b \\ & & 1 & & b \end{matrix} ] , ∀ a, b ∈ ℂ\}, the Jordan forms:

(a) 3J_{1}(a) ⊕ 2J_{1}(b) if a ≠ b,

(b) J_{1}(a) ⊕ 2J_{2}(a) if a = b;

4. P_{π_{4}} = \{[ \begin{matrix}a \\ & a \\ & & a \\ & & 1 & & b \\ & & & 1 & c \end{matrix} ] , ∀ a, b, c ∈ ℂ\}, the Jordan form depends on the

roots of the polynomial x^{2} −cx−b - the characteristic polynomial of the block

\left[{b \atop 1 c}\right]:

(a) 2 simple roots λ_{1}, λ_{2}, λ_{i} ≠ a - 3J_{1}(a) ⊕ J_{1}(λ_{1}) ⊕ J_{1}(λ_{2}),

(b) 2 simple roots λ_{1}, λ_{2}, λ_{1} = a, λ_{2} ≠ a - 2J_{1}(a) ⊕ J_{2}(a) ⊕ J_{1}(λ_{2}),

(c) 1 double root λ_{1} ≠ a - 3J_{1}(a) ⊕ J_{2}(λ_{1}),

(d) 1 double root λ_{1} = a - 2J_{1}(a) ⊕ J_{3}(a),

5. P_{π_{5}} = \{[ \begin{matrix}& & a \\ & & & a \\ 1 & & b \\ & 1 & & b \\ & & & 1 & c \end{matrix} ] , ∀ a, b, c ∈ ℂ\}, the Jordan form depends on the

roots of the polynomial x^{2} − bx − a - the characteristic polynomial of the

block \left[{a \atop 1 b}\right]:

(a) 2 simple roots λ_{1}, λ_{2}, λ_{i} ≠ c - J_{1}(c) ⊕ 2J_{1}(λ_{1}) ⊕ 2J_{1}(λ_{2}),

(b) 2 simple roots λ_{1}, λ_{2}, λ_{1} = c, λ_{2} ≠ c - J_{1}(c) ⊕ J_{2}(c) ⊕ 2J_{1}(λ_{2}),

(c) 1 double root λ_{1} ≠ c - 2J_{2}(λ_{1}) ⊕ J_{1}(c),

(d) 1 double root λ_{1} = c - J_{2}(c) ⊕ J_{3}(c),

6. P_{π_{6}} = \{[ \begin{matrix}a \\ & a \\ & 1 & & & b \\ & & 1 & & c \\ & & & 1 & d \end{matrix} ] , ∀ a, b, c, d ∈ ℂ\}, the Jordan form depends on the

roots of the polynomial x^{3} − dx^{2} − cx − b - the characteristic polynomial of

the block [ \begin{matrix}& & b \\ 1 & & c \\ & 1 & d \end{matrix} ]:

(a) 3 simple roots λ_{1}, λ_{2}, λ_{3}, λ_{i} ≠ a - 2J_{1}(a) ⊕ J_{1}(λ_{1}) ⊕ J_{1}(λ_{2}) ⊕ J_{1}(λ_{3}),

(b) 3 simple roots λ_{1}, λ_{2}, λ_{3}, λ_{1} = a, λ_{2,3} ≠ a - J_{1}(a) ⊕ J_{2}(a) ⊕ J_{1}(λ_{2}) ⊕ J_{1}(λ_{3}),

(c) 1 double root λ_{1} ≠ a and 1 simple root λ_{3} ≠ a - 2J_{1}(a)⊕J_{2}(λ_{1})⊕J_{1}(λ_{3}),

(d) 1 double root λ_{1} = a and 1 simple root λ_{3} ≠ a - J_{1}(a) ⊕ J_{3}(a) ⊕ J_{1}(λ_{3}),

(e) 1 double root λ_{1} ≠ a and 1 simple root λ_{3} = a - J_{1}(a) ⊕ J_{2}(a) ⊕ J_{2}(λ_{1}),

(f) 1 triple root λ_{1} ≠ a - 2J_{1}(a) ⊕ J_{3}(λ_{1}),

(g) 1 triple root λ_{1} = a - J_{1}(a) ⊕ J_{4}(a);

7. P_{π_{7}} = \{[ \begin{matrix}& & & & a \\ 1 & & & & b \\ & 1 & & & c \\ & & 1 & & d \\ & & & 1 & e \end{matrix} ] , ∀ a, b, c, d, e ∈ ℂ\}, the Jordan form depends on

the roots of the characteristic polynomial of the matrix x^{5} −ex^{4} −dx^{3} −cx^{2} − bx − a:

(a) 5 simple roots λ_{i}, i ∈ {1, ..., 5} - ⨁_{i=1}^{5} J_{1}(λ_{i}),

(b) 1 double root λ_{1} and 3 simple roots λ_{i}, i ∈ {3, ..., 5} - J_{2}(λ_{1}) ⊕ J_{1}(λ_{3}) ⊕ J_{1}(λ_{4}) ⊕ J_{1}(λ_{5}),

(c) 2 double roots λ_{1}, λ_{3} and 1 simple root λ_{5} - J_{2}(λ_{1}) ⊕ J_{2}(λ_{3}) ⊕ J_{1}(λ_{5}),

(d) 1 triple root λ_{1} and 2 simple roots λ_{4}, λ_{5} - J_{3}(λ_{1}) ⊕ J_{1}(λ_{4}) ⊕ J_{1}(λ_{5}),

(e) 1 triple root λ_{1} and 1 double root λ_{4} - J_{3}(λ_{1}) ⊕ J_{2}(λ_{4}),

(f) 1 root of order 4 λ_{1} and 1 single root λ_{5} - J_{4}(λ_{1}) ⊕ J_{1}(λ_{5}),

(g) 1 root of order 5 λ_{1} - J_{5}(λ_{1}).

REFERENCES

[1] V.I.Arnold (Ed.) Arnold’s problems. Springer, 2004.

[2] R.A.Lippert, G.Strang The Jordan forms of AB and BA. Electronic Journal of Linear Algebra, 18:281–288, 2009.