GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 2, 1996, 141-152

CRITERIA OF UNITARY EQUIVALENCE OF HERMITIAN OPERATORS WITH A DEGENERATE SPECTRUM

I. LOMIDZE

Abstract. Nonimprovable, in general, estimates of the number of necessary and sufficient conditions for two Hermitian operators to be unitarily equaivalent in a unitary space are obtained when the multiplicities of eigenvalues of operators can be more than 1. The explicit form of these conditions is given. In the Appendix the concept of conditionally functionally independent functions is given and the corresponding necessary and sufficient conditions are presented.

Let P, Q be the operators from a unitary n-dimensional space 𝕌^{n} in 𝕌^{n}, and P, Q be the matrices of these operators in some orthonormal basis. Description of a system of invariants of these matrices which enables one to find out whether the given operators are unitarily equivalent is the classical problem of the theory of invariants (see, e.g., [1, §2.2], [2] and the references cited therein). In the author’s paper [3] it is shown that two matricess P, Q ∈ M_{n}(ℂ) are unitarily equivalent iff the following conditions are fulfilled:

tr {P_{+}^{l}P_{−}P_{+}^{m}P_{−}^{2}} = tr {Q_{+}^{l}Q_{−}Q_{+}^{m}Q_{−}^{2}}, 0 ≤ l ≤ m ≤ n − 1,

tr {P_{+}^{l}} = tr {Q_{+}^{l}}, 1 ≤ l ≤ n, (1)

where A_{+} (A_{−}) denotes the Hermitian (skew-Hermitian) part of the matrix A ∈ M_{n}(ℂ):

A_{±} = (A ± A^{∗})/2.

Formulas (1) contain n(n + 3)/2 of complex (but only n^{2} + 1 of real) conditions, and all these conditions are independent if no additional restrictions are imposed on the entries of the matrices P_{±}, Q_{±}. However, if such restrictions are imposed, in particular, if some eigenvalues of the operator P_{+} have multiplicity ≥ 2, then not all of conditions (1) are independent [3].

141 There arises a problem of finding a minimal number of necessary and sufficient conditions of type (1) for the matrices P, Q ∈ M_{n}(ℂ) to be unitarily equivalent.

1991 Mathematics Subject Classification. 15A72, 13A50, 15A15, 15A54.

Key words and phrases. Invariance of operators, functional independence.

1072-947X/96/0300-0141$09.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1996 Plenum Publishing Corporation

Let the operators P, Q : 𝕌^{n} → 𝕌^{n} be Hermitian and have eigenvalues whose multiplicities can be more than 1. In this paper we derive an estimate (not improvable in the general case) of the number of necessary and sufficient conditions for unitary equivalence of the Hermitian matrices P, Q ∈ M_{n}(ℂ) corresponding to these operators and state these conditions in explicit form. It is shown that the number of independent conditions 𝔫 ≤ n and all cases are found when the equality 𝔫 = n is fulfilled.

Let n eigenvalues of the Hermitian operator P : 𝕌^{n} → 𝕌^{n} form the multiset [4, §3.4]

P = \{p_{i}^{r_{i}}|i = \overline{1, m}\}, ∑_{i=1}^{m} r_{i} = n, p_{i} ≠ p_{j} for i ≠ j, r_{i} ∈ ℕ.

Let us denote

t_{k}(P) = tr P^{k} = ∑_{i=1}^{m} r_{i}p_{i}^{k}, k = 0, 1, 2, …,  (2)

and let

\begin{matrix}T_{k}(P) = [T_{ij}(P)]_{1≤i,j≤k} = [t_{i+j−2}(P)] ∈ M_{k}(ℝ), \\ D_{k}(P) = det T_{k}(P), k ∈ ℕ, \end{matrix}  (3)

stand for Hankel matrices and their determinants. It is obvious that D_{1}(P) = t_{0}(P) = n.

Since all eigenvalues of the operator P : 𝕌^{n} → 𝕌^{n} are real, the rank of matrix T_{k}(P) for sufficiently large k is equal to the signature of this matrix and to the number m of various eigenvalues [5, Ch. 16, §9]:

rank T_{k}(P) = m, k ≥ m.  (4)

Lemma. The following formulas hold for the determinants (3):

D_{k}(P) =
= ∑_{1≤i_{1}<···<i_{k}≤m} {r_{i_{1}} ⋯ r_{i_{k}} ∏_{1≤j<l≤k} (p_{i_{j}} − p_{i_{l}})^{2}} > 0, 1 ≤ k ≤ m. (5)

Proof. By (2) we have

D_{k}(P) = det AB = det [ ∑_{j=1}^{m} a_{ij}(P)b_{jl}(P)],  (6)

where A ∈ M_{k,m}, B ∈ M_{m,k} stand for

A = [a_{ij}(P)] = [r_{j}p_{j}^{i−1}], B = [b_{jl}(P)] = [p_{j}^{i−1}], 1 ≤ j ≤ m, 1 ≤ i, l ≤ k.

(As usual, M_{k,l} = M_{k,l}(ℝ) denotes the set of real matrices of dimension k × l, k, l ∈ ℕ, and M_{k} = M_{k,k}.)

Applying the Cauchy–Binet formula to (6), we obtain (5).

Corollary. The power series (Newtonian sums) t_{k}(P) for k ≥ 2m are rational sums of the variables t_{0}(P), …, t_{2m−1}(P).

Proof. It follows from (4) that

D (\begin{matrix}1, & ⋯, & m, & m + 1 \\ 1, & ⋯, & m, & m + k \end{matrix}) = 0, k ∈ ℕ,

for all minors surrounding the minor D_{m}(P) > 0, which implies

t_{2m+k−1}(P)D_{m}(P) =
= ∑_{i,j=1}^{m} t_{m+k+j−1}(P)t_{m+i−1}(P)A_{m}\binom{j}{i}(P), k ∈ ℕ,  (7)

where A_{m}\binom{j}{i}(P) is the algebraic complement of the element t_{j+i−2}(P) in D_{m}(P) depending evidently only on t_{k}(P), k = \overline{0, 2m − 2}. By the induction with respect to k we obtain the desired result from (8).

Theorem 1. The mapping

\{t_{k}(P)|k = \overline{0, 2m − 1}\} ↦ \{p_{i}^{r_{i}}|i = \overline{1, m}\}  (8)

is bijective.

Proof. The injectivity of mapping (8) follows from (2). Let us prove the surjectivity. It is obvious that the Hermitian operator P : 𝕌^{n} → 𝕌^{n} having m ≤ n various eigenvalues p_{1}, …, p_{m} satisfies the operator identity

P^{m} − ∑_{k=1}^{m} c_{k}P^{m−k} = 0  (9)

with c_{k} = (−1)^{k} ∑_{1≤i_{1}<···<i_{k}≤m} p_{i_{1}}p_{i_{2}} ⋯ p_{i_{k}}, k = \overline{1, m}.

By multiplying (9) by P^{0}, …, P^{m−1} and calculating traces we obtain the following system of equations for coefficients c_{k}:

∑_{k=1}^{m} c_{k}t_{m−k+i}(P) = t_{m+i}(P), i = \overline{0, m − 1},  (10)

whose determinant is (−1)^{m(m−1)/2}D_{m}(P) ≠ 0. After calculating the unknowns c_{k}, k = \overline{1, m}, from (10) we find p_{1}, p_{2}, …, p_{m} (p_{i} ≠ p_{j} for i ≠ j) as roots of the polynomial z^{m} − ∑_{k=1}^{m} c_{k}z^{m−k} = 0.

By substituting the found values of p_{i} in the first m equations of system (2) and recalling that the determinant of this system (with respect to unknowns r_{i}) is

det [p_{i}^{k−1}]_{1}^{m} = ∏_{1≤k<j≤m}(p_{j} − p_{k}) ≠ 0,

we obtain r_{1}, r_{2}, …, r_{m}.

Note that to calculate the Jacobian of mapping (8) it is sufficient to know the determinant of the Vandermonde 2-multiple matrix [6]

J = \frac{∂(t_{0}, t_{1}, …, t_{2m−1})}{∂(p_{1}, r_{1}, …, p_{m}, r_{m})} = ( ∏_{i=1}^{m} r_{i}) det(p_{1}, …, p_{m}; 2) =

= m!( ∏_{i=1}^{m} r_{i}) ∏_{1≤k<j≤m}(p_{j} − p_{k})^{2^{2}} ≠ 0.

Since all r_{i} ≥ 1, we can define the diagonal matrix R = diag{r_{1}^{−1}, …, r_{m}^{−1}}. Following (2) and (3), we use the notation

t_{k}(R) = tr R^{k}, D_{k}(R) = det T_{k}(R), k ∈ ℕ (t_{0}(R) = m). (11)

Theorem 2. The mapping

\{ tr P^{k}|k = \overline{0, 2m − 1}\} ↦ \{ tr R^{k}|k ∈ ℕ\}

is injective and all tr R^{k}, k ∈ ℕ, are the rational functions of the arguments t_{0}(P), …, t_{2m−1}(P).

Proof. For the sake of brevity we denote t_{k} = tr P^{k}, T_{k} = T_{k}(P), t̃_{k} = tr R^{k}. From the first m equations of (2) we find

r_{i} = \frac{det R^{(i)}}{det[p_{i}^{k−1}]_{1}^{m}} , i = \overline{1, m},  (12)

where R^{(i)} = [R_{kl}^{(i)}] ∈ M_{m} is the matrix whose kth row has the form

[p_{1}^{k−1}, …, p_{i−1}^{k−1}, t_{k−1}, p_{i+1}^{k−1}, …, p_{m}^{k−1}], k = \overline{1, m}.

Due to (12) we obtain

r_{i}^{2} = \frac{det[r_{l}R_{kl}^{(i)}] det[R_{kl}^{(i)}]}{det[r_{j}p_{j}^{k−1}] det[p_{j}^{k−1}]}, i = \overline{1, m}.

If we cancel r_{i} in both sides and multiply the determinants in the right-hand side according to the “row by row” rule, we obtain

r_{i} = \frac{det[t_{k+j−2} + t_{k−1}t_{j−1} − r_{i}p_{i}^{k+j−2}]_{1}^{m}}{det[t_{k+j−2}]_{1}^{m}} , i = \overline{1, m}.  (13)

To simplify (13) let us introduce one-column matrices 0 = [0] ∈ M_{m,1}, t = [t_{k−1}] ∈ M_{m,1}, p_{i} ∈ [p_{i}^{k−1}] ∈ M_{m,1} (i = \overline{1, m}). Use the following matrix identity for (m + 2) × (m + 2) block-matrices (i = \overline{1, m}):

[\begin{matrix}T_{m} & t & −r_{i}p_{i} \\ −t′ & +1 & 0 \\ −p′_{i} & 0 & +1 \end{matrix}] [\begin{matrix}E_{m} & 0 & 0 \\ t′ & +1 & 0 \\ p′_{i} & 0 & +1 \end{matrix}] = [\begin{matrix}T_{m} + tt′ − r_{i}p_{i}p′_{i} & t & −r_{i}p_{i} \\ 0′ & +1 & 0 \\ 0′ & 0 & +1 \end{matrix}]

(here E_{m} ∈ M_{m} is the identity matrix and ′ denotes transposition), which implies

det [T_{m} + tt′ − r_{i}p_{i}p′_{i}] =

= det [\begin{matrix}T_{m} & t & −r_{i}p_{i} \\ −t′ & +1 & 0 \\ −p′_{i} & 0 & +1 \end{matrix}] = det [\begin{matrix}T_{m} & 0 & −r_{i}p_{i} \\ 0′ & 1 + t_{0} & −r_{i} \\ −p′_{i} & +1 & +1 \end{matrix}] .

The latter matrix is obtained from the previous one by adding the first and the (m + 1)th rows and then subtracting the first colum from the (m + 1)th one. Expanding the result with respect to the elements of the (m + 1)th row we get for each i = \overline{1, m} that

det [T_{m} + tt′ − r_{i}p_{i}p′_{i}] = (1 + t_{0} + r_{i})D_{m} + (1 + t_{0}) det [\begin{matrix}T_{m} & −r_{i}p_{i} \\ −p′_{i} & 0 \end{matrix}] .

Substitution into (13) and simplification give

r_{i} det [\begin{matrix}T_{m} & p_{i} \\ −p′_{i} & 0 \end{matrix}] = D_{m}, i = \overline{1, m}.

Hence we obtain

r_{i}^{−1} = ∑_{j=0}^{2m−2} p_{i}^{j}c_{j},  (14)

where the coefficients c_{j}, j = \overline{0, 2m − 2}, are expressed rationally through t_{0}, …, t_{2m−1} as follows:

c_{j} = D_{m}^{−1} ∑_{l=0}^{j} A_{m}\binom{j − l + 1}{l + 1}(P), j = \overline{0, 2m − 2}.  (15)

By virtue of (2), (7), (11), and (15) we find from (14) that

t̃_{k−1} = ∑_{i=1}^{m} r_{i}^{−k+1} = ∑_{j=0}^{2k(m−1)} t_{j} ∑_{j_{1}+···+j_{k}=j} c_{j_{1}} ⋯ c_{j_{k}} = f_{k}(t_{0}, …, t_{2m−1}),

where f_{k}(t_{0}, …, t_{2m−1}) is a rational function of its arguments, k ∈ ℕ.

Let the primary specification (see [7]) of multiset P be also the multiset

\{r_{i}| i = \overline{1, m}\} = \{q_{i}^{s_{i}}| i = \overline{1, l}\}, q_{i} ≠ q_{j} for i ≠ j, q_{i}, s_{i} ∈ ℕ, (16)

where

∑_{i=1}^{l} q_{i}s_{i} = n, ∑_{i=1}^{l} s_{i} = m.  (17)

Denote by p_{ij}, j = \overline{1, s_{i}}, the eigenvalues of operator P : 𝕌^{n} → 𝕌^{n} each having the multiplicity equal to q_{i}, i = \overline{1, l}. Without loss of generality, multiset P will be assumed to be ordered so that

p_{ij} < p_{i,j+1}, j = \overline{1, s_{i}}, q_{i} < q_{i+1}, i = \overline{1, l − 1}.  (18)

In these notation we have

t_{k}(P) = ∑_{i=1}^{l} (q_{i} ∑_{j=1}^{s_{i}} p_{ij}^{k}), t_{k}(R) = ∑_{i=1}^{l} s_{i}q_{i}^{−k}, k = 0, 1, 2, ….

Following the lemma, the determinants in (11) satisfy the conditions D_{k}(R) > 0 for k ≤ l and D_{k}(R) = 0 for k ≥ l + 1. Hence on account of Theorem 2 we obtain m − l conditions satisfied by values t_{0}(P), …, t_{2m−1}(P). Thus the set \{t_{k}(P)| k = \overline{0, 2m − 1}\} contains at most m + l − 1 = 𝔫(P) independent elements.

Remark. In terms of the partition theory formulas (17) imply that multiset (16) is the partitioning of the number n, which is a dimension of the space 𝕌^{n}, and the rank of this partitioning is m. By the notation of [4] we have

(q_{1}^{s_{1}}, …, q_{l}^{s_{l}}) ⊢ n, (s_{1}, …, s_{l}) ⊢ m.

Following the Ramsay theorem (see [7]), 𝔫(P) is the greatest number each of whose partitioning into l parts

(𝔫_{1}, …, 𝔫_{l}) ⊢ 𝔫(P)

contains at least one part having the property

𝔫_{i} ≤ s_{i}, 1 ≤ i ≤ l.

Proposition. For 𝔫(P) we have the estimate

𝔫(P) = l + ∑_{i=1}^{l} s_{i} − 1 ≤ n,

the equality being fulfilled if either all eigenvalues of the operator P : 𝕌^{n} → 𝕌^{n} are simple:

l = 1; q_{1} = 1, s_{1} = n; (1^{n}) ⊢ n,

or if one eigenvalue has multiplicity 2 while the rest of the eigenvalues are simple:

l = 2; q_{1} = 1, s_{1} = n − 2; q_{2} = 2, s_{2} = 1; (1^{n−2}, 2) ⊢ n.

Proof. Formulas (17) imply

n = ∑_{i=1}^{l}(q_{i} − 1)(s_{i} − 1) + ∑_{i=1}^{l} s_{i} + ∑_{i=1}^{l}(q_{i} − 1) =

= ∑_{i=1}^{l}(q_{i} − 1)(s_{i} − 1) + ∑_{i=1}^{l} s_{i} + ∑_{i=1}^{l}(q_{i} − i) + l(l − 1)/2.

Hence on account of the inequalities s_{i} ≥ 1, q_{i} ≥ i we have the estimate

n ≥ ∑_{i=1}^{l} s_{i} + l(l − 1)/2,

in which the equality holds if q_{i} = i, i = \overline{1, l}, and s_{2} = ⋯ = s_{l} = 1. Note that l(l − 1)/2 ≥ l − 1 with equality for l = 1, 2.

Example. Let the multiset of eigenvalues of the operator P have the form:

(a) \{p_{1j}, p_{i}^{q_{i}}| j = \overline{1, s_{1}}, i = \overline{2, l}\}, i.e., the operator P has s_{1} simple eigen-

values; then (17) implies l(l + 1)/2 ≤ n − s_{1} + 1 and
𝔫(P) = s_{1} + 2l − 2, s_{1} ≤ 𝔫(P) ≤ \sqrt{9 + 8(n − s_{1})} + s_{1} − 3;

(b) {p_{1}, p_{2}^{n−1}}; then l = 2; q_{1} = 1, s_{1} = 1, q_{2} = n − 1, s_{2} = 1; 𝔫(P) = 3;

(c) {p_{1}^{n}}; then l = 1; s_{1} = 1, q_{1} = n, 𝔫(P) = 1.

Let us construct an ordered set of invariants of an operator P (of a matrix P) of the form

I(P) = \{t_{k}(P)| k = \overline{1, 𝔫(P)}, 𝔫(P) = l + ∑_{i=1}^{l} s_{i} − 1\}.  (19)

Theorem 3. For the Hermitian operator P : 𝕌^{n} → 𝕌^{n} the set of invariants (19) is complete and all elements of this set are functionally independent as functions of the independent variables (18).

Proof. We introduce the notation

ϕ_{ij} = r_{i,j+1} − r_{i1}, j = \overline{1, s_{i} − 1}, i = \overline{1, l},

where r_{ij} is the multiplicity of the eigenvalues p_{ij}, j = \overline{1, s_{i}}, i = \overline{1, l}. By virtue of the theorem on conditional functional independence (see the Appendix) it is sufficient to show that the functions \{t_{k}(P)| k = \overline{1, 𝔫(P)}\} are conditionally functionally independent in the presence of constraints

ϕ_{ij} = 0, j = \overline{1, s_{i} − 1}, i = \overline{1, l}, t_{0}(P) = ∑_{i=1}^{l} q_{i}s_{i} = n.

Calculate the Jacobian

J̃ = \frac{∂(t_{0}, t_{1}, …, t_{𝔫(P)}, ϕ_{11} …, ϕ_{ls_{l}})}{∂(p_{11}, r_{11} …, p_{ls_{l}}, r_{ls_{l}})} .

Taking (3) into account and performing some simple calculations, we find

J̃ = ( ∏_{i=1}^{l} q_{i}^{s_{i}}) det [ũ_{1}| … |ũ_{l}],  (20)

where

ũ_{i} = [u_{kj}^{(i)}]_{\begin{matrix}0≤k≤𝔫(P) \\ 1≤j≤s_{i} \end{matrix}} = [ ∑_{j=1}^{s_{i}} p_{ij}^{k}, (p_{i1}^{k})′, …, (p_{is_{i}}^{k})′]_{0≤k≤𝔫(P)}, i = \overline{1, l}.

Here (p_{ij}^{k})′ = kp_{ij}^{k−1} = (∂/∂p_{ij})p_{ij}^{k}.

Applying induction with respect to l, let us show that the determinant in the right-hand side of (20) is not identically zero. Indeed, for l = 1 we have

det[ũ_{1}] = det [ ∑_{j=1}^{s_{1}} p_{1j}^{k}, (p_{11}^{k})′, …, (p_{1s_{1}}^{k})′]_{0≤k≤s_{1}} =

= s_{1}(s_{1})! ∏_{1≤l<j≤s_{1}}(p_{1j} − p_{1l}) ≠ 0.

Assume that l ≥ 2 and det[ũ_{1}| ⋯ |ũ_{l−1}] ≠ 0. After expanding det[ũ_{1}| ⋯ |ũ_{l}] with respect to the last s_{l} + 1 rows we obtain

det [ũ_{1}| ⋯ |ũ_{l}] = det [ũ_{1}| ⋯ |ũ_{l−1}] ×
× det [ ∑_{j=1}^{s_{l}} p_{lj}^{k−1}, (p_{l1}^{k−1})′, …, (p_{ls_{l}}^{k−1})′]_{𝔫(P)−s_{l}≤k≤𝔫(P)} + ⋯, (21)

where the points denote the terms of lower powers with respect to the variables \{p_{lj}| j = \overline{1, s_{l}}\}. Treating det[∑_{j=1}^{s_{l}} p_{lj}^{k−1}, p_{l1}^{k−1′}, …, p_{ls_{l}}^{k−1′}] = P (p_{l1}) as the polynomial of the variable p_{l1}, we obtain

P (p_{l1}) = det [p_{l1}^{k−1}, (p_{l1}^{k−1})′, …, (p_{ls_{l}}^{k−1})′]_{𝔫(P)−s_{l}≤k≤𝔫(P)} + ⋯ =

= p_{l1}^{2𝔫(P)−4} det [(p_{l2}^{k−1})′, …, (p_{ls_{l}}^{k−1})′]_{𝔫(P)−s_{l}≤k≤𝔫(P)−2} + ⋯ =

= p_{l1}^{2𝔫(P)−4} \frac{(𝔫(P) − 3)!}{(𝔫(P) − s_{l} − 2)!} ∏_{j=2}^{s_{l}} p_{lj}^{𝔫(P)−s_{l}−2} ∏_{2≤t<j≤s_{l}}(p_{lj} − p_{lt}) + · · · .

After substituting this result into (21) and taking into account the assumption of induction we find that det[ũ_{1}| ⋯ |ũ_{l}] is the polynomial of p_{l1} of the power

2𝔫(P) − 4 = 2(l + ∑_{i=1}^{l} s_{i} − 1) − 4 ≥ 4l − 6 ≥ 2

with a higher coefficient which is not identically zero. Therefore J̃ ≠ 0.

Corollary. Two given Hermitian matrices P, Q ∈ M_{n}(ℂ) are unitarily equivalent iff 𝔫(P) real equalities

I(P) = I(Q)  (22)

are fulfilled, where I(P) is determined by (19).

Proof. The necessity of conditions (22) is obvious. The sufficiency follows from the fact that the mapping

I(P) ↦ \{p_{ij}, q_{i}| j = \overline{1, s_{i}}, i = \overline{1, l}\}

is injective by virtue of Theorem 3.

Example.

(a) The Hermitian matrix P ∈ M_{n}(ℂ) is proportional with a coefficient a ∈ ℝ to E_{n} iff D_{2}(P) = t_{0}t_{2} − t_{1}^{2} = 0, or, which is the same,

tr P^{2} = (tr P)^{2}/n.  (23)

It is obvious that P = n^{−1}(tr P)E_{n}.

(b) The Hermitian matrix P ∈ M_{n}(ℂ) is equal to zero iff condition (23) is fulfilled and tr P = 0.

Remark. If some additional restrictions are imposed on elements of the Hermitian matrix P, i.e., if not all of these elements are independent (e.g., for tr P = 0), then the complete system of invariants of the operator P : 𝕌^{n} → 𝕌^{n} contains less than 𝔫(P) independent elements.

Appendix

Assume that n, m ∈ ℕ, we are given n + m differentiable functions of n + m variables x_{1}, …, x_{n}, y_{1} …, y_{m}

f_{i} : ℝ^{n+m} → ℝ, i = \overline{1, n + m},  (A.1)

such that the Jacobian J_{1} differs from zero:

J_{1} = \frac{∂(f_{1}, …, f_{n+m})}{∂(x_{1}, …, x_{n}, y_{1}, …, y_{m})} ≠ 0

and the variables x_{1}, …, x_{n}, y_{1}, …, y_{m} satisfy m constraints

ϕ_{j}(x_{1}, …, x_{n}, y_{1}, …, y_{m}) = 0, j = \overline{1, m},  (A.2)

where each of functions ϕ_{j} : ℝ^{n+m} → ℝ, j = \overline{1, m}, is differentiable with respect to n + m arguments x_{1}, …, x_{n}, y_{1}, …, y_{m} and

J_{2} = \frac{∂(ϕ_{1}, …, ϕ_{m})}{∂(y_{1}, …, y_{m})} ≠ 0.

As known, constraint equations (A.2) determine explicit functions

y_{j} = y_{j}(x_{1}, …, x_{n}), j = \overline{1, m},

and functions (A.1) become composite functions of the variables x_{1}, …, x_{n}:

f_{i} = f_{i}{(x_{1}, …, x_{n}, y_{1}(x_{1}, …, x_{n}), …, y_{m}(x_{1}, …, x_{n})}, (A.3)
i = \overline{1, n + m}.

Definition. The functions in (A.1) will be called conditionally functionally dependent (independent) in the presence of constraints (A.2) provided that the corresponding composite functions (A.3) are functionally dependent (independent), i.e., if the Jacobian det[f_{i,k}] is equal to (different from) zero, where

f_{i,k} = ∂_{x_{k}}f_{i} + ∑_{l=1}^{m} (∂_{y_{l}}f_{i})(∂_{x_{k}}y_{l}), i, k = \overline{1, n},  (A.4)

and ∂_{x_{k}}y_{l} are uniquely determined by the system of equations

0 = ∂_{x_{k}}ϕ_{j} + ∑_{l=1}^{m} (∂_{y_{l}}ϕ_{j})(∂_{x_{k}}y_{l}), k = \overline{1, n}, j = \overline{1, m}.  (A.5)

Theorem. In the presence of constraints (A.2), only those n functions from the set \{f_{i}|i = \overline{1, n + m}\} will be conditionally functionally dependent for which the Jacobian

J = \frac{∂(f_{1}, …, f_{n}, ϕ_{1}, …, ϕ_{m})}{∂(x_{1}, …, x_{n}, y_{1}, …, y_{m})} = det [\begin{matrix}∂_{x_{k}}f_{i} & | & ∂_{y_{l}}f_{i} \\ — — & | & — — \\ ∂_{x_{k}}ϕ_{i} & | & ∂_{y_{l}}ϕ_{j} \end{matrix}]

is equal to zero.

For J ≠ 0 the functions f_{1}, … f_{n} are conditionally functionally independent.

Proof. For the sake of brevity we denote the Jacobi matrices by

F = [\begin{matrix}F_{11} & F_{12} \\ F_{21} & F_{22} \end{matrix}] ∈ M_{n+m},

Φ = [Φ_{1} | Φ_{2}] ∈ M_{m,n+m}, Y = [∂_{x_{k}}y_{l}] ∈ M_{n,m},

where

F_{11} = [∂_{x_{k}}f_{i}] ∈ M_{n}, F_{12} = [∂_{y_{l}}f_{i}] ∈ M_{n,m}, i = \overline{1, n},

F_{21} = [∂_{x_{k}}f_{i}] ∈ M_{m,n}, F_{22} = [∂_{y_{l}}f_{i}] ∈ M_{m}, i = \overline{n + 1, n + m},

Φ_{1} = [∂_{x_{k}}ϕ_{j}] ∈ M_{m,n}, Φ_{2} = [∂_{y_{l}}ϕ_{j}] ∈ M_{m}, j = \overline{1, m}

(k = \overline{1, n}, l = \overline{1, m}).

In this notation formulas (A.4) and (A.5) have the form

[f_{i,k}] = F_{11} + F_{12}Y, −Φ_{1} = Φ_{2}Y

and

J_{1} = det F ≠ 0, J_{2} = det Φ_{2} ≠ 0.

Hence

det [f_{i,k}] = det [F_{11} − F_{12}Φ_{2}^{−1}Φ_{1}].

Applying now the known identity for a block matrix determinant (see, e.g., [5, Ch.2, §5]) we obtain

det [f_{i,k}] = J_{2}^{−1} det [\begin{matrix}F_{11} & F_{12} \\ Φ_{1} & Φ_{2} \end{matrix}] = J/J_{2}.

Thus the condition det[f_{i,k}] = 0 is equivalent to the condition J = 0.

Corollary. If ϕ_{j} = f_{n+j}, j = \overline{1, m}, and J_{1} ≠ 0 then

det [f_{i,k}] = J_{1}/J_{2} ≠ 0.

References

1. R. A. Horn and Ch. R. Johnson, Matrix analysis. Cambridge University Press, Cambridge, 1986.

2. V. L. Popov and E. B. Vinberg, Algebraic geometry IV. Invariant theory. (Encyclopaedia of Mathematical Sciences, v. 55). Springer-Verlag, Berlin, 1994.

3. I. R. Lomidze, Criteria of unitary and orthogonal equivalence. (Russian) Bull. Acad. Sci. Georgia 141(1991), No. 3,481–483.

4. G. E. Andrews, Theory of partitions. Encyclopaedia of Mathematics and its Applications, v. 2. Addison Wesley, Reading, 1976.

5. F. R. Gantmakher, The theory of matrices. (Russian) Nauka, Moscow, 1988.

6. I. Lomidze, On some generalizations of the Vandermonde matrix and their relations with the Euler beta-function. Georgian Math. J. 1(1994), No. 4, 405–417.

7. V. I. Baranov and B. S. Stechkin, Extreme combinatorial problems and their applications. (Russian) Nauka, Moscow, 1989.

(Received 17.05.1994)

Author’s address: Faculty of Physics I. Javakhishvili Tbilisi State University 3, I. Chavchavadze Ave., Tbilisi 380028 Republic of Georgia