V. P. ZYLYOV.

Generalization of the Concept and Properties of a Matrix Composed of Minors of Another Matrix

1. In generalizing the theorems relating to a matrix composed of the minors of another matrix, we extend the concept of a compound matrix to such an extent that a further increase in the scope of the concept seems impossible, if only one matrix is taken as the basis.

In deriving the theorems we shall use the concept of the rank of matrices and their reduction to normal form by means of elementary transformations.

By an elementary transformation of a matrix is meant one of the following operations: a) transposition of rows, b) multiplication of all elements of a row by one and the same non-zero number, and c) addition to the elements of one row of the corresponding elements of a parallel row.

We shall call the elementary-normal form of a matrix, to which it can be brought by means of elementary transformations alone, i.e. without changing its rank, such a form of the matrix in which all its elements are equal to zero, with the exception of \(\rho\) elements equal to one and situated at the intersections of the \(\rho\) first rows with the corresponding \(\rho\) first columns. The number \(\rho\) is equal to the rank of the matrix*).

1. Let us take the matrix

\[ \left\| \begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{1m}\\ a_{21} & a_{22} & \ldots & a_{2m}\\ \ldots & \ldots & \ldots & \ldots\\ a_{n1} & a_{n2} & \ldots & a_{nm} \end{array} \right\| \tag{\(\Phi\)} \]

Let us form all combinations of rows taken \(v_1\) at a time, then \(v_2\) at a time, and so on, finally \(v_p\) at a time, where \(v_1+v_2+\cdots+v_p<n\).

To each of the combinations taken \(v_1\) at a time we shall successively adjoin each of the combinations taken \(v_2\) at a time that contains not a single one of the rows of the combination to which it is adjoined; we shall obtain \(C_n^{v_1}\cdot C_{n-v_1}^{v_2}\) pairs of unions.

* See D. Grave, Elements of Higher Algebra, §§ 50–59. Kiev, 1914.

To each of these pairs we successively adjoin each of the combinations of \(v_3\) that contains not a single row of the pair to which the adjunction is being made; we obtain
\[ C_n^{r_1}\cdot C_{n-r_1}^{r_2}\cdot C_{n-r_1-r_2}^{v_3} \]
combinations.

We continue the process until all the numbers \(v_i\) have been exhausted; we obtain
\[ C_n^{r_1}\cdot C_{n-r_1}^{r_2}\cdots C_{n-r_1-\cdots-r_{p-1}}^{r_p} = \binom{r_1 r_2\cdots r_p}{n} = k_n \]
combinations of rows.

Each of these combinations consists of \(p\) groups: the first contains \(v_1\) rows, the second \(v_2\), the third \(v_3\), and so on, the \(p\)-th contains \(v_p\) rows; one of these compound combinations will be denoted by
\[ [(i_{11}, i_{12},\ldots i_{1r_1}),\ (i_{21},\ldots i_{2r_2}),\ldots (i_{p1},\ldots i_{pr_p})], \]
where the \(i\)’s denote the numbers of the rows; of this combination we shall say that it consists of the first, second, …, \(p\)-th group of rows.

We proceed in exactly the same way with the columns; we obtain
\[ C_m^{r_1}\cdots C_{m-r_1-\cdots-r_{p-1}}^{r_p}=k_m \]
analogous combinations; one of the combinations will be denoted by
\[ [(j_{11}, j_{12},\ldots j_{1r_1}),\ldots (j_{p1},\ldots j_{pr_p})]; \]
this combination consists of the first, second, …, \(p\)-th group of columns.

Arrange in some order all the combinations of rows, then all the combinations of columns; call them the first, second, etc., combinations
\[ \left\{ \begin{array}{c} \text{of rows}\\ \text{of columns}. \end{array} \right. \]
We form the matrix
\[ (\Theta)\qquad \left\| \begin{array}{cccc} A_{11} & A_{12} & \cdots & A_{1k_m}\\ A_{21} & A_{22} & \cdots & A_{2k_m}\\ \cdots & \cdots & \cdots & \cdots\\ A_{k_n1} & A_{k_n2} & \cdots & A_{k_nk_m} \end{array} \right\| \]
so composed from the matrix \((\Phi)\) that \(A_{ij}\) is equal to the product of \(p\) determinants standing at the intersection of the \(i\)-th combination of rows with the \(j\)-th combination of columns; namely, the first of these determinants consists of the elements of the matrix \((\Phi)\) lying at the intersection of the \(v_1\) rows of the first group of the \(i\)-th row-combination with the \(v_1\) columns of the first group of the \(j\)-th column-combination; analogously the second determinant is obtained, and so on, finally, the \(p\)-th.

We shall call the determinants whose product constitutes \(A_{ij}\), in accordance with what has been set forth, the first, second, …, \(p\)-th determi-

by the elements \(A_{ij}\). We shall call the matrix \((\Theta)\) a compound matrix of \((\Phi)\), and \((\Phi)\), in relation to \((\Theta)\), the base.

Let us explain the definition of a compound matrix by the following schematic example:

\[ \begin{array}{c} \\[-1em] \begin{array}{ccccccc} & & v_1\ \text{column} & & v_2\ \text{column} & & v_3\ \text{column} \end{array} \\ \begin{array}{c|c|c|c|c|c|c|c} & & & & & & & \\ v_1\ \text{row} & & L_1 & & & & & \\ & & & & & & & \\ (\Phi) & & & & & & & \\ & h & & & L_2 & & & h \\ v_2\ \text{row} & & & & L_2 & & & \\ & g & & & & & & g \\ & & & & & & & \\ v_3\ \text{row} & & & & & & L_3 & \\ & & & & & & & \end{array} \end{array} \]

In the example \(p=3\). Here is represented the intersection of one of the combinations of rows,—for example, the third,—of the matrix \((\Phi)\) with one of the combinations of columns, for example—the fifth.

A combination of \(\left\{\begin{array}{c}\text{rows}\\ \text{columns}\end{array}\right.\) consists of three groups: the first contains \(v_1\), the second \(v_2\), the third \(v_3\) \(\left\{\begin{array}{c}\text{rows}\\ \text{columns}\end{array}\right.\).

The first group of a combination of rows intersects with the first group of a combination of columns in the determinant, denoted briefly by \(L_1\); the second group of a combination of rows intersects with the second group of a combination of columns in the determinant \(L_2\), and, finally, the third groups of the combinations intersect in \(L_3\). Then the element of the compound matrix is
\[ A_{ij}=L_1 L_2 L_3; \]
where \(L_1, L_2, L_3\) are respectively the first, second, and third determinants.

The intersection of the \(i\)-th combination of rows with the \(1, 2, \ldots, k_m\) combinations of columns gives the \(i\)-th row of the matrix \((\Theta)\), and the intersection of the \(j\)-th combination of columns with the first, second, etc., \(h_n\), combinations of rows gives the \(j\)-th column of the matrix \((\Theta)\).

Remark. For what follows it is important to know in what way the elements of a row (column) of the matrix \((\Phi)\) enter into the rows of the matrix \((\Theta)\).

Without loss of generality, we may turn, in order to clarify this circumstance, to our schematic example. Two rows have been marked: the row \(gg\), whose elements do not enter into \(A_{35}\), and the row \(hh\), whose elements do enter into \(A_{35}\); namely, part of the row \(hh\) gives one of the rows of the second determinant \(A_{35}\). If, in this way, we consider the intersection of the third combination of rows with all combinations of columns, then we observe that each element of the third row of the compound matrix contains, as a factor, a determinant—in our example, the second one—all elements of one of whose rows belong to the row \(hh\) of the matrix \((\Phi)\).

Consequently, if we multiply all elements of the row \(hh\) by the number \(a\), then by this very fact we multiply all rows of the matrix \((\Theta)\) that include the elements of the row \(hh\) by the same number.

It is not difficult to see that the number of rows of the matrix \((\Theta)\) that include the elements of any row of the matrix \((\Phi)\) is equal to

\[ k_n-k_{n-1}=C_n^{i_1}C_{n-r_1}^{i_2}\cdots C_{\,n-r_1-\cdots-r_{p-1}}^{i_p} - C_{n-1}^{i_1}C_{\,n-1-r_1}^{i_2}\cdots C_{\,n-1-r_1-\cdots-r_{p-1}}^{i_p}. \]

Indeed, the number of all combinations of rows is equal to \(k_n\); the number of combinations not including one of the rows of the matrix \((\Phi)\) is, evidently,

\[ k_{n-1}=C_{n-1}^{i_1}\cdots C_{\,n-1-r_1-\cdots-r_{p-1}}^{i_p}. \]

Consequently, the number of combinations including the row is equal to \(k_n-k_{n-1}\), and therefore in the matrix \((\Theta)\) there will be \(k_n-k_{n-1}\) rows with elements of some one row of \((\Phi)\).

Hence, finally, we obtain the important conclusion: if the elements of a row of the matrix \((\Phi)\) are multiplied by \(a\), then \(k_n-k_{n-1}\) rows of the matrix \((\Theta)\) are multiplied by \(a\).

Everything said about rows is also applicable to columns.

3. Let us prove several theorems.

We shall call an elementary transformation of a matrix a matrix obtained from the first by means of elementary transformations.

Theorem I. The compound matrix of an elementary transformation of the matrix \((\Phi)\) is an elementary transformation of the compound matrix of \((\Phi)\).

For the proof we shall consider separately each of the three elementary transformations.

The first elementary transformation of the matrix \((\Phi)\), for example—transposition of rows—disturbs only the order of the row combinations indicated in the definition of the compound matrix: the 1st, 2nd, ..., \(k_n\) combina-

...will pass respectively into \(i_1, i_2,\ldots,i_{k_n}\). The compound matrix of the new matrix will, consequently, differ from the compound matrix of the original one only in that its 1st, 2nd, etc. rows will pass respectively into \(i_1\), \(i_2\)-nd, etc. But the substitution
\[ \begin{pmatrix} 1 & 2 & \ldots & k_n\\ i_1 & i_2 & \ldots & i_{k_n} \end{pmatrix} \]
can be replaced by a series of transpositions; therefore, for the first transformation the theorem is valid.

The second elementary transformation \((\Phi)\), for example, multiplication by \(a\) of some column of the matrix, is equivalent, as has already been shown in the remark at the end of § 2, to multiplication by \(a\) of \(k_{n-1}\) columns \((\Theta)\).

Thus, for the second transformation the theorem is valid.

Finally, the third elementary transformation, for example, adding to the elements of the 3rd row of the matrix \((\Phi)\) the corresponding elements of the 8th row, is equivalent, in the compound matrix, to adding rows to rows; indeed, those rows of the matrix \((\Theta)\) which formerly contained elements of the 3rd row \((\Phi)\) will now contain the elements of the 3rd row, altered by the addition of the 8th. Expanding the determinants \(L\) into the sum of two determinants, we find that in the matrix \((\Theta)\), to the rows containing elements of the 3rd row \((\Phi)\), there have been added rows containing elements of the 8th.

Thus, for the 3rd transformation the theorem is also valid. A conclusion from this theorem:

The rank of the compound matrix of \((\Phi)\) is equal to the rank of the compound matrix of an elementary transformation of the matrix \((\Phi)\).

Theorem II. If the rank of the matrix \((\Phi)\) is \(\rho\), then the rank of the compound matrix \((\Theta)\) is equal to
\[ k_\rho = C_{\rho}^{r_1}\, C_{\rho-r_1}^{r_2} \cdots C_{\rho-r_1\ldots-r_{\rho-1}}^{r_\rho}. \]

On the basis of the conclusion from Theorem I, it is evidently sufficient to determine the rank of the compound matrix from the elementary-normal form of the matrix \((\Phi)\).

The elementary-normal form of the matrix \((\Phi)\) is the following
\[ \left| \begin{array}{cccccccc} 1 & 0 & 0 & \ldots & 0 & 0 & \ldots & 0\\ 0 & 1 & 0 & \ldots & 0 & 0 & \ldots & 0\\ 0 & 0 & 1 & \ldots & 0 & 0 & \ldots & 0\\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots\\ 0 & 0 & 0 & \ldots & 1 & 0 & \ldots & 0\\ 0 & 0 & 0 & \ldots & 0 & 0 & \ldots & 0\\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots\\ 0 & 0 & 0 & \ldots & 0 & 0 & \ldots & 0 \end{array} \right| \qquad (\Phi_1 \]

Here the number of elements equal to one is \(\rho\), which is equal to the rank of the matrices \((\Phi)\) and \((\Phi_1)\).

We form from \((\Phi_1)\) a composite as follows: first we compose \(k_\rho\) combinations of the rows and columns of the determinant matrix

\[ \delta = \begin{vmatrix} 1 & 0 & \cdot & 0 \\ 0 & 1 & \cdot & 0 \\ \cdot & \cdot & \cdot & \cdot \\ 0 & 0 & \cdot & 1 \end{vmatrix}, \]

and then we shall take combinations containing zero rows.

Combinations of rows containing zero rows obviously give zero rows in the matrix \((\Theta_1)\).

We arrange the \(k_\rho\) combinations of rows and the \(k_\rho\) combinations of columns of the matrix \(\delta\) so that between the combinations of rows on the one hand and of columns on the other there is a complete correspondence, i.e., if, for example, the 5th combination of rows has the form \([(2, . . 3); (4, . . 7, . . 8); \ldots]\), then the 5th combination of columns must be constructed from the columns in exactly the same way, i.e. must have the form \([(2, . . 3); (4, . . 7, . . 8); \ldots]\). These notations fully correspond to the general notation of § 2, the rows being arranged in increasing order of their numbers.

We shall show that the matrix \((\Theta_1)\) will have the same form as the matrix \((\Phi_1)\), with the difference that the number of ones will be equal to

\[ k_\rho = C_{\rho}^{p_1} \ldots C_{\rho-r_1-\cdots-r_{p-1}}^{p_p} = \operatorname{rank}(\Theta_1)=\operatorname{rank}(\Theta). \]

Indeed, take any combination of rows of the matrix \((\Phi_1)\), the \(s\)-th, for example,

\[ \tag{1} [(i_{11}, i_{12}, . . i_{1r_1});\ (i_{21}, i_{22}, . . i_{2r_2});\ (i_{p1}, . . i_{pr_p})] \]

and the \(t\)-th combination of columns,

\[ \tag{2} [(j_{11}, . . j_{1r_1});\ (j_{21}, j_{22}, . . j_{2r_2});\ \ldots (j_{p1}, . . j_{pr_p})]. \]

If \(s=t\le k_\rho\), then (2) will be exactly the same as (1), and each determinant of intersection is equal to 1, i.e. \(A_{ss}=1\).

If \(s\ne t\), then at least one of the groups of the combination (2), for example the second, differs from the corresponding group of the combination (1). This in turn indicates that at least one of the numbers \(j_{21}, j_{22}, . . j_{2r_2}\), for example \(j_{22}\), is not equal to any of the numbers \((i_{21}, . . i_{2r_2})\); then the determinant standing at the intersection of the rows \((i_{21}, i_{22}, . . i_{2r_2})\) with the columns \((j_{21}, j_{22}, . . j_{2r_2})\) is equal to zero. Indeed, one of its columns consists only of zeros: the \(j_{22}\)-th column, intersecting

with \(i_{21}, i_{22}, \ldots i_{2r_2}\), gives only zeros, because in the determinant \(\delta\) all elements, except the principal ones, are equal to \(0\); thus, for \(s\ne t\), \(A_{st}=0\); consequently, \((\Theta_1)\) has the form \((\Phi)\), and the number of units is equal to \(k_\rho\); by this we have proved our theorem.

Corollary.

\(1^\circ\). The compound matrix of the normal form of the matrix \((\Phi)\) is the normal form of the compound of \((\Phi)\)*).

\(2^\circ\). Passing from the general case to the particular one, let us put \(p\) equal to one, i.e. in forming the compound matrix let us combine the rows and the columns only by \(v_1\). In this case, already known to mathematicians, each element of the matrix consists of one determinant, and not of a product of \(p\) determinants, as in the general case.

Applying the general theorem, we obtain:

The rank of the simple compound matrix is equal to \(C_\rho^{v_1}\), if \(\rho\) is the rank of the original one.

1. Let us apply what has been set forth to square matrices, in particular—to the determinants expressed by them.

From the matrix of a determinant of order \(n\), \(D\), we form the compound matrix of the determinant, which we shall denote by \(\Delta\).

Theorem III. The compound determinant is equal to a power of the original one, whose exponent is equal to

\[ C_n^{r_1} C_{n-r_1}^{r_2}\ldots C_{n-r_1-\cdots-r_{p-1}}^{r_p} - C_{n-1}^{r_1} C_{n-1-r_1}^{r_2}\ldots C_{n-1-r_1-\cdots-r_{p-1}}^{r_p}, \]

i.e.

\[ \Delta = D^{\,k_n-k_{n-1}}. \]

If \(D=0\), then also \(\Delta=0\), since its order is \(k_n\), while its rank is equal to \(k_\rho\), and for \(\rho<n\) we have \(k_\rho<k_n\).

Consequently, the theorem need be proved for \(D\ne0\).

We reduce the matrix of the determinant \(D\) to the elementary-normal form \(D_1\). Only the second of the elementary transformations changes the magnitude of the determinant, namely—if \(a_1, a_2,\ldots a_t\) are the multipliers by which the rows of \(D\) had to be multiplied in order to obtain \(D_1\), then

\[ D_1 = a_1 a_2\ldots a_t D. \tag{1} \]

On the basis of the corollary of the first theorem of the second, the matrix \(\Delta\) will thereby pass into the elementary-normal form \(\Delta_1\), and, on the basis of the remark at the end of § 2,

\[ \Delta_1 = a_1^{\,k_n-k_{n-1}}\cdot a_2^{\,k_n-k_{n-1}}\ldots a_t^{\,k_n-k_{n-1}}\cdot \Delta . \tag{2} \]

*) Here the concept of the normal form of a matrix is not restricted only to the elementary-normal form.

Raising both parts of (1) to the power with exponent \(k_n-k_{n-1}\), we obtain

\[ \mathrm{D}_1^{\,k_n-k_{n-1}}=(a_1a_2\ldots a_t)^{\,k_n-k_{n-1}}\cdot \mathrm{D}^{\,k_n-k_{n-1}}, \]

and since

\[ \Delta_1=\mathrm{D}_1=1, \]

then, by comparing these equalities with the preceding ones, we find

\[ \Delta=\mathrm{D}^{\,k_n-k_{n-1}}. \]

Consequences:

\(1^\circ\). For \(p=1\) we obtain the well-known theorem (Spottiswoode–Sylvester)

\[ \tag{3} \Delta=\mathrm{D}^{\,C_n^{v_1}-C_{n-1}^{v_1}}=\mathrm{D}^{\,C_{n-1}^{v_1-1}}. \]

The new derivation of this truth differs considerably from the known ones; moreover, \(\mathrm{D}\) is not assumed here to be only a determinant of general form.

\(2^\circ\). Still more special is the case when, for \(p=1\), we have \(v_1=n-1\); in this case \(\Delta\) becomes the reciprocal determinant to \(\mathrm{D}\), and equality (3) gives:

\[ \tag{4} \Delta=\mathrm{D}^{\,C_{n-1}^{n-2}}=\mathrm{D}^{\,n-1}. \]

We note that the theorem on the magnitude of the reciprocal determinant may be obtained independently of the theorem on the product of determinants, since, although we spoke of the product of determinants, we did not use the theorem on multiplying them.

1. For each of the elements of the determinant \(\Delta\), one can find in \(\mathrm{D}\) the corresponding adjoint (algebraic complement); the determinant obtained from \(\Delta\) by replacing its elements by the just-mentioned adjoints will be denoted by \(\delta\).

Applying the method of ordinary reasoning, while using the generalized theorem of Laplace on the expansion of determinants, we obtain two theorems:

\[ \mathrm{IV})\quad \delta=\mathrm{D}^{\,k_n-(k_n-k_{n-1})}=\mathrm{D}^{\,k_{n-1}}, \]

\[ \mathrm{V})\quad f_q=\mathrm{D}^{\,q-k_{n-1}}\cdot g_{q_1} \]

which is a generalization of Franke’s theorem*).

Here \(f_q\) is a minor of the \(q\)-th order of the determinant \(\Delta\), \(g_{q_1}\) is a minor of the \((k_n-q)\)-th order of the determinant \(\delta\), complementary to the minor corresponding to \(f_q\); for \(p=1\) we obtain Franke’s formula, and for \(p=1\) and \(v_1=n-1\) the formula for a minor of the reciprocal determinant.

*) See Pascal, §§ 22–24.

1. In conclusion we shall give two examples of the application of what has been set forth.

1st example. The Capelli–Garbieri theorem*): “if the rank of the matrix \((\Phi)\) is equal to \(\rho\), then the determinants of order \(\rho\) contained in any \(\rho\) rows are proportional to the corresponding determinants contained in other \(\rho\) parallel rows.” If it is applied to the compound matrix \((\Theta)\) formed from \((\Phi)\), of rank \(K_\rho\), then, obviously, we obtain a greater generalization of the theorem, since the matrix \((\Phi)\) is a special case of \((\Theta)\).

2nd example: using the method of § 4, we shall derive the theorem on the multiplication of determinants. Suppose first that the determinants being multiplied are nonzero.

We have two determinants

\[ (1)\ |a_{ij}| \quad \text{and} \quad (2)\ |b_{ij}| \]

Let us form from them a new determinant \((3)\ |c_{ij}|\) such that

\[ c_{ij}=a_{i1}b_{j1}+a_{i2}b_{j2}+\cdots+a_{in}b_{jn}, \]

and let us prove that \(|c_{ij}|=|a_{ij}|\cdot|b_{ij}|\).

We reduce the matrices of the determinants \(a_{ij}\) and \(b_{ij}\) to normal form. Only the second elementary transformation changes their value, introducing multipliers.

The elementary-normal forms of the corresponding matrices give determinants equal to unity

\[ (4)\quad 1=m_1m_2\cdots m_t\cdot |a_{ij}| \]

\[ (5)\quad 1=q_1q_2\cdots q_t\cdot |b_{ij}|, \]

where \(m\) and \(q\) are the multipliers by which the rows of the matrices \((1)\) and \((2)\) had to be multiplied. Multiplication of a row of one of the determinants \((1)\) and \((2)\), as is easily seen from the form of \(c_{ij}\), gives multiplication of the corresponding row of the determinant \(|c_{ij}|\) by the same number. Moreover, the determinant \((3)\), formed for the elementary-normal forms of the matrices \((1)\) and \((2)\), is, obviously, the elementary-normal form \(c_{ij}\) and is equal to \(1\); hence

\[ (6)\qquad 1=m_1m_2\cdots m_t\cdot q_1q_2\cdots q_t\cdot |c_{ij}|. \]

Multiplying \((4)\) and \((5)\) and comparing with \((6)\), we find

\[ |c_{ij}|=|a_{ij}|\cdot|b_{ij}|. \]

If one or both determinants are equal to zero, then from the form of the elementary-normal form \(c_{ij}\) we see that also \(|c_{ij}|=0\).

Tomsk, January 7, 1915.
V. Zylev.

) Capelli–Garbieri, Analisi algebraica*, p. 398; 1886. The text is in Pascal–Leitzmann.