UDC 512.864

MATHEMATICS

L. A. KALUZHNIN, H. HAVIDI

GEOMETRIC THEORY OF UNITARY EQUIVALENCE OF MATRICES

(Presented by Academician A. I. Mal’tsev on XI 22, 1965)

1. Let \(\mathfrak A=\{A_1,A_2,\ldots,A_m\}\) and \(\mathfrak B=\{B_1,B_2,\ldots,B_m\}\) be two sequences of length \(m\) of matrices of order \(n\) over the field of complex numbers. \(\mathfrak A\) and \(\mathfrak B\) are called unitarily equivalent if there exists a unitary matrix \(S\) such that \(B_i=S^{-1}A_iS\) for all \(i=1,2,\ldots,m\). For the case when \(m=1\) and the matrices \(A_1\) and \(B_1\) are normal, a necessary and sufficient condition for unitary equivalence (u.e.) is the coincidence of the eigenvalues. In recent years numerous papers have been devoted to the general case \((^1\!-\!^7)\). In the present note this question is treated in terms of \(n\)-dimensional unitary geometry. We indicate a procedure for the successive computation of numerical invariants for \(\mathfrak A\) and \(\mathfrak B\), whose coincidence is necessary and sufficient for their u.e. Our results are analogous to those obtained in \((^7)\), but the geometric interpretation, in our view, is more transparent and admits far-reaching generalizations.

2. One may restrict oneself to the case when all the matrices \(A_i\) and \(B_i\) are normal. Indeed, every matrix \(A\) admits a unique representation in the form \(A=A^{(1)}+iA^{(2)}\), where \(A^{(1)}\) and \(A^{(2)}\) are Hermitian and, consequently, normal. For the u.e. of the sequences \(\mathfrak A\) and \(\mathfrak B\) it is necessary and sufficient that the sequences of length \(2m\), \(\{A_i^{j}\}\), \(\{B_i^{(j)}\}\), \(i=1,2,\ldots,m;\ j=1,2\), be u.e. Thus, we assume from the outset that \(\mathfrak A\) and \(\mathfrak B\) consist of normal matrices.

3. To each matrix \(A_i\) (respectively \(B_i\)) there corresponds a set of eigenvalues and an orthogonal decomposition of the unitary space \(V\) of dimension \(n\) into a sum of admissible subspaces whose dimensions are equal to the multiplicities of the corresponding eigenvalues.

Thus, the sequence \(\mathfrak A\) determines \(m\) orthogonal decompositions

\[ \begin{aligned} V&=U_{11}\perp U_{12}\perp\cdots\perp U_{1r_1},\\ V&=U_{21}\perp U_{22}\perp\cdots\perp U_{2r_2},\\ &\cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\\ V&=U_{m1}\perp U_{m2}\perp\cdots\perp U_{mr_m} \end{aligned} \tag{1} \]

and, analogously, for the sequence \(\mathfrak B\),

\[ \begin{aligned} V&=T_{11}\perp T_{12}\perp\cdots\perp T_{1s_1},\\ V&=T_{21}\perp T_{22}\perp\cdots\perp T_{2s_2},\\ &\cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\\ V&=T_{m1}\perp T_{m2}\perp\cdots\perp T_{ms_m}. \end{aligned} \tag{2} \]

A series of orthogonal decompositions of the type (1) or (2) will be called a configuration; the orthogonal summands \(U_{ij}\) and \(T_{ij}\), its components.

Theorem 1. For the unitary equivalence of \(\mathfrak A\) and \(\mathfrak B\) it is necessary and sufficient: a) that the eigenvalues of the matrices \(A_i\) and \(B_i\) coincide (including their multiplicities), and (provided condition a) is fulfilled) b) that the configurations (1) and (2) be u.e.

Condition b) means the existence of a unitary operator \(A\) on \(V\), carrying simultaneously all components \(U_{ij}\) onto the corresponding compo-

elements \(T_{ij}\). In what follows we shall deal with u.e. configurations in which \(r_j=s_j\), \(j=1,2,\ldots,m\).

1. Invariants for u.e. configurations (1) and (2), along with the dimensions of the corresponding components, are the “angles” between pairs of corresponding components \((U_{ij},U_{lk})\) and \((T_{ij},T_{ik})\).

By the angle \(\angle(U,W)\) of subspaces \(U\) and \(W\) of the space \(V\) one means the following object (see, for example, (8)). Let \(\pi_U\) and \(\pi_W\) be the projectors of \(V\) onto \(U\) and \(W\); \(\pi_U^W\) (respectively \(\pi_W^U\)) is the restriction of \(\pi_U\) (respectively \(\pi_W\)) to \(W\) (respectively to \(U\)).

Definition. The angle \(\angle(U,W)\) between the subspaces \(U\) and \(W\) is the operator on \(U\) equal to \(\pi_U^W\pi_W^U\) (first \(\pi_W^U\), then \(\pi_U^W\)). Similarly, \(\angle(W,U)\) is the operator \(\pi_W^U\pi_U^W\) on \(W\).

It turns out that \(\angle(U,W)\) and \(\angle(W,U)\) are nonnegative self-adjoint (and, consequently, normal!) operators on \(U\) and \(W\). Moreover: a) for the eigenvalues \(\lambda_i\) of the angle \(\angle(U,W)\) one has \(0\le \lambda_i\le 1\); the multiplicity of the eigenvalue 1 is equal to the dimension of \(U\cap W\); the multiplicity of 0 is equal to the dimension of \(U\cap W^\perp\) (\(W^\perp\) is the orthogonal complement of \(W\)) and similarly for the eigenvalues \(\lambda_i'\) of the angle \(\angle(W,U)\); b) the eigenvalues \(\lambda_i\) and \(\lambda_i'\) coincide pairwise, including multiplicities, except possibly for the multiplicities of the eigenvalue 0.

Let \(d(\lambda_i)\) be the multiplicity of the eigenvalue \(\lambda_i\). Renumber the \(\lambda_i\) in increasing order:
\[ 0=\lambda_0<\lambda_1<\cdots<\lambda_{r-1}=1 \]
and similarly for \(\lambda_i'\). The expression
\[ \begin{pmatrix} \lambda_0'=0, & \lambda_0=0, & \lambda_1,\ldots,\lambda_{r-1}\\ d(\lambda_0') & d(\lambda_0) & d(\lambda_1),\ldots,d(\lambda_{r-1}) \end{pmatrix} \]
will be called the metric characteristic of the pair of subspaces \(U,W\).

Theorem 2. For two u.e. pairs \((U,W)\) and \((U',W')\) of subspaces it is necessary and sufficient that the metric characteristics of these pairs coincide.

The proof of Theorem 2 is contained, for example, in (8).

For unitary equivalence of configurations (1) and (2), it is necessary that the metric characteristics of all possible angles \(\angle(U_{ij},U_{lk})\) from (1) coincide with the characteristics of the angles \(\angle(T_{ij},T_{lk})\) from (2) for the corresponding pairs of subspaces. But these numerous conditions, generally speaking, are not sufficient. Indeed, for the component \(U_{ij}\), the angles \(\angle(U_{ij},U_{lk})\)—for all possible \(U_{lk}\)—besides the metric characteristics, also determine orthogonal decompositions of \(U_{ij}\) into the eigenspaces of the angles \(\angle(U_{ij},U_{lk})\), and as new invariants there arise, for example, the angles between the eigenspaces of \(\angle(U_{ij},U_{lk})\) and \(\angle(U_{ij},U_{rs})\) in \(U_{ij}\). The problem of u.e. configurations, in a certain sense, reduces to an analogous one, but for subspaces of smaller dimension. In the next section we shall carry out such a reduction in more detail.

1. The angle \(\angle(U,W)\) between subspaces \(U,W\subset V\) will be called scalar if it is a scalar linear operator \(\lambda E_U\) (where \(E_U\) is the identity operator in \(U\)). We note that if \(\angle(U,W)\) is scalar and the dimensions of the subspaces \(U\) and \(W\) are different, then \(\lambda=0\) and, consequently, \(U\) and \(W\) are orthogonal.

Suppose that in configurations (1) and (2) the metric characteristics of pairs of corresponding components coincide, and that for some pair of components \(U_{ij}\) and \(U_{lk}\) (and, consequently, also for the corresponding components \(T_{ij}\) and \(T_{lk}\)) the angles \(\angle(U_{ij},U_{lk})\) and \(\angle(U_{lk},U_{ij})\) are not simultaneously scalar. Then, decomposing in configuration (1) (and respectively in (2)) the spaces \(U_{ij}\) and \(U_{lk}\) into the orthogonal sum of eigenspaces corresponding to the given angles, we obtain two new configurations \((1')\) and \((2')\), which we shall call a refinement of configurations (1) and (2), or, more precisely, a refinement with respect to the indices \(ij\) and \(lk\).

Let us note—it is not difficult to prove—that the newly arising components will form scalar angles with one another. It is also not difficult to prove that the condensed configurations \((1')\) and \((2')\) will be u.e. if and only if the original configurations \((1)\) and \((2)\) are u.e. Carrying out condensations successively, after a finite number of steps we arrive at configurations, say \((1^*)\) and \((2^*)\), such that the angles between any two components are scalar. The configurations \((1^*)\) and \((2^*)\) will be called reduced. The passage to reduced configurations is not uniquely determined, since, generally speaking, it will depend on which pairs of indices are condensed at each step. But it can be made consistent.

Theorem 3. Configurations \((1)\) and \((2)\) are u.e. if and only if:
a) all metric characteristics between the corresponding pairs of components coincide at every step of condensation, and b) the reduced configurations \((1^*)\) and \((2^*)\) are u.e.

Suppose that \((1)\) and \((2)\) are already reduced configurations. It turns out that in reduced configurations the following simple situation occurs.

Theorem 4. Let, for every \(d \le n\), \(V_{dj}\) be the orthogonal sum of all components of dimension \(d\) in the \(j\)-th row of the configuration; then
\[ V_{d1}=V_{d2}=\cdots=V_{dm}. \]

Theorems 3 and 4 reduce the problem of u.e. of reduced configurations to the u.e. of configurations of the form
\[ \begin{aligned} V&=U_{11}\perp U_{12}\perp\cdots\perp U_{1k},\\ V&=U_{21}\perp U_{22}\perp\cdots\perp U_{2k},\\ &\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ . \\ V&=U_{m1}\perp U_{m2}\perp\cdots\perp U_{mk}, \end{aligned} \tag{3} \]
\[ \begin{aligned} V&=T_{11}\perp T_{12}\perp\cdots\perp T_{1k},\\ V&=T_{21}\perp T_{22}\perp\cdots\perp T_{2k},\\ &\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ . \\ V&=T_{m1}\perp T_{m2}\perp\cdots\perp T_{mk}, \end{aligned} \tag{4} \]
where all components \(U_{ik}\) and \(T_{ik}\) have one and the same dimension \(d\); \(n=dk\); all angles between components are scalar. Configurations of this kind will be called special.

It turns out that for the u.e. of two special configurations \((3)\) and \((4)\) it is necessary and sufficient that: a) the scalar angles between pairs of corresponding components coincide:
\[ \angle(U_{ij},U_{lk})=\angle(T_{ij},T_{lk}); \]
b) there be u.e. of two subsequently defined sequences of unitary matrices of order \(d<n\). Thus, finally, the u.e. of a sequence of matrices of order \(n\) is reduced to an analogous problem for matrices of order \(d<n\). For \(d=1\), the corresponding sequences are sets of complex numbers of modulus equal to 1. For u.e. they must coincide.

In the following two sections we deal with special configurations \((3)\) and \((4)\).

1. Two components \(U_{ij}\) and \(U_{lk}\) will be called adjacent, and the adjacency relation will be denoted by \(U_{ij}CU_{lk}\), if and only if
   \[ \angle(U_{ij},U_{lk})\ne 0. \]
   By \(S\) we shall denote the transitive closure of the relation \(C\): \(U_{ij}SU_{lk}\) if and only if there exists a sequence of pairs of indices
   \[ ij=i_0j_0,\ i_1j_1,\ldots,\ i_rj_r=lk, \]
   such that
   \[ \Psi_{ij,lk}:\quad U_{ij}CU_{i_1j_1}\wedge U_{i_1j_1}CU_{i_2j_2}\wedge\cdots\wedge U_{i_{r-1}j_{r-1}}CU_{lk}; \tag{5} \]
   \(\Psi_{ij,lk}\) will be called a path from \(U_{ij}\) to \(U_{lk}\). We can and shall restrict ourselves only to such paths in which no pair occurs more than once. There are only finitely many such paths in the configuration.

\(S\) is an equivalence relation, and if the number \(p\) of equivalence classes with respect to \(S\) is \(p>1\), then the rows of the configurations split into \(p\) orthogonal blocks of components from one class, such that any two components from different blocks (different rows) are orthogonal. For unitary equivalence of configurations (3) and (4) it is necessary and sufficient that: a) the scalar angles coincide; b) the decomposition into blocks be the same; c) there be unitary equivalence of the configurations consisting of components belonging to the same equivalence class with respect to \(S\).

1. If \(p=1\), then the special configuration is called connected. We shall indicate necessary and sufficient conditions for unitary equivalence of connected special configurations. To every path \(\Psi_{ij,lk}\) from \(U_{ij}\) to \(U_{lk}\) (see (5)) we assign the following operator from \(U_{ij}\) to \(U_{lk}\):

\[ \Pi(\Psi_{ij,lk})= \frac{1}{\sqrt{\lambda(ii_1,i_1j_1)\cdots\lambda(i_{r-1}j_{r-1},lk)}} \pi^{U_{lk}}_{U_{i_{r-1}j_{r-1}}}\cdots \pi^{U_{i_1j_1}}_{U_{ij}}, \tag{6} \]

where \(\lambda(mn,qt)E_{mn}=\angle(U_{mn},U_{qt})\), and the square root is positive. \(\Pi(\Psi_{ij,lk})\) carries every orthonormal basis in \(U_{ij}\) into a uniquely determined orthonormal basis in \(U_{lk}\).

Now choose in \(U_{11}\) some basis \(\mathbf e_1,\mathbf e_2,\ldots,\mathbf e_d\), and let, for some \(U_{lk}\), \(lk\ne 11\), \(\Psi_{ij,lk}\) run through a nonempty (in view of connectedness!) and finite set of paths from \(U_{11}\) to \(U_{lk}\). If \(\Psi'_{11,lk}\) and \(\Psi''_{11,lk}\) are two such paths, then between the bases \(\Pi(\Psi'_{11,lk})(\mathbf e_1,\mathbf e_2,\ldots,\mathbf e_d)\) and \(\Pi(\Psi''_{11,lk})(\mathbf e_1,\mathbf e_2,\ldots,\mathbf e_d)\) there is a certain unitary transition matrix of order \(d\). We denote it by
\((\Psi'_{11,lk}\to\Psi''_{11,lk})(\mathbf e_1,\mathbf e_2,\ldots,\mathbf e_d)\). Generally speaking (examples show that this is indeed so), it depends on the initial basis \(\mathbf e_1,\mathbf e_2,\ldots,\mathbf e_d\), but if one passes to another orthonormal basis in \(U_{11}\) by means of some transition matrix \(L\), then \((\Psi'_{11,lk}\to\Psi''_{11,lk})(\mathbf e_1,\mathbf e_2,\ldots,\mathbf e_d)\) is transformed by means of \(L\). From this it is already not difficult to obtain the final result.

Theorem 5. For unitary equivalence of connected special configurations (3) and (4), it is necessary and sufficient that the following conditions be satisfied: a) coincidence of the scalar angles between pairs of corresponding components; b) for arbitrarily chosen orthonormal bases \(\mathbf e_1,\mathbf e_2,\ldots,\mathbf e_d\) in \(U_{11}\) and \(\mathbf f_1,\mathbf f_2,\ldots,\mathbf f_d\) in \(T_{11}\), and for all \(lk\ne 11\), simultaneous unitary equivalence of the transition matrices
\((\Psi'_{11,lk}\to\Psi''_{11,lk})(\mathbf e_1,\mathbf e_2,\ldots,\mathbf e_d)\) (for (3)) with the corresponding matrices
\((\Phi'_{11,lk}\to\Phi''_{11,lk})(\mathbf f_1,\mathbf f_2,\ldots,\mathbf f_d)\) (for (4)) for all possible pairs of paths from \(U_{11}\) to \(U_{lk}\) and, respectively, from \(T_{11}\) to \(T_{lk}\).

Remark 1. The obtained systems of invariants for unitary equivalence are highly redundant.

Remark 2. The reduction in item 5 (coalescing) and the choice of \(U_{11}\) and \(T_{11}\) in item 7 as the initial point of the paths depend on the arbitrariness of the numbering of the components.

Kyiv State University
named after T. G. Shevchenko

Received
17 XI 1965

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