V. P. ZYLYOV.

On Two Properties of Related Determinants.

1. In May 1914, Professor V. L. Nekrasov proposed to me that I generalize the results of his interesting work: “On one property of related determinants”*).

I succeeded in obtaining broad generalizations, expressed by two theorems. Both theorems had been discovered earlier, one—by Capelli and Garbieri**), the other in a particular form—by A. Cayley***) and in general—by E. von Weber****).

I note that E. von Weber erroneously attributes A. Cayley’s theorem to L. Kronecker, while W. H. Metzler is mistaken in thinking that the generalization of A. Cayley’s theorem was made by him only in 1914.

The generalization of Prof. V. L. Nekrasov’s theorems, as well as a new proof of the Capelli–Garbieri theorem, I venture to present to the attention of readers.

2. According to V. L. Nekrasov, determinants are called related determinants if they are selected from one and the same matrix. The rank, or characteristic, of a matrix, according to Frobenius, is called the greatest order of its minors that are not equal to zero.

Take a matrix consisting of \(n\) rows and \(m > n\) columns; if its rank \(p < n\), then it is called vanishing.

Capelli–Garbieri Theorem:

If the rank of a matrix is equal to \(p\), then related determinants contained in \(p\) rows (columns) are proportional to the corresponding determinants contained in any \(p\) rows (columns).

*) Izvestiya of the Tomsk Technological Institute of Emperor Nicholas II, vol. 32, 1913.

**) A. Capelli e G. Garbieri. Corso di analisi algebrica, 1, Padoue 1886, p. 398.

***) Cambr. math. I. 4 (184³/₅), p. 119.

****) E. von Weber. Vorlesungen über das Pfaff’sche Problem, 1900, p. 13.

Our proof is based on the following theorem, well known from the theory of determinants:

If a determinant of order \((p+1)\) is equal to zero, then its minors contained in \(p\) rows (columns) are proportional to the corresponding determinants contained in any \(p\) rows (columns). We shall make use of a schematic drawing.

Drawing 1.

\((\Phi)\)

Schematic determinant diagram with the indicated minors \(A_{ik}\), \(A_{it}\), \(A_{qk}\), and \(A_{qt}\).

Let us denote by \(A_{ik}\) the determinant of order \(p\); by \(A_{it}\), the determinant contained in the same rows; and by \(A_{qk}\), the determinant contained in the same columns as \(A_{ik}\); finally, by \(A_{ik+1}\) we shall call the determinant obtained from \(A_{ik}\) by shifting it along its rows by 1 column, and by \(A_{i+1k}\), the determinant obtained from \(A_{ik}\) by shifting it along its columns by 1 row; the rows of \(A_{qk}\) and the columns of \(A_{it}\) may partially coincide, respectively, with the rows and columns of the determinant \(A_{ik}\).

It is required to prove that

\[ \frac{A_{ik}}{A_{qk}}=\frac{A_{it}}{A_{qt}}. \]

If the rank \(p=n\), then, obviously, \(A_{ik}=A_{qk}\), and \(A_{it}=A_{qt}\), since the number of all rows is equal to \(n\) and, consequently, the rows of the determinants \(A_{qk}\) and \(A_{qt}\) coincide with the rows of the determinants \(A_{ik}\) and \(A_{it}\); then

\[ \frac{A_{ik}}{A_{qk}}=1=\frac{A_{it}}{A_{qt}}; \]

but this case is of no interest. We now pass to the proof of the theorem for \(p<n\), i.e. for a vanishing matrix.

Let (see Drawing 2) the rectangles \(AB\), \(ED\), \(FH\), \(CJ\) denote respectively the determinants \(A_{ik}\), \(A_{i\,k+1}\), \(A_{i+1\,k}\), \(A_{i+1\,k+1}\).

Drawing 2.

Then \(AJ\) denotes a determinant of order \((p+1)\) which, since the rank of the matrix is \(p\), is equal to zero. Consequently, on the basis of the theorem cited,

\[ \frac{A_{ik}}{A_{i+1\,k}}=\frac{A_{i\,k+1}}{A_{i+1\,k+1}}. \]

Since the argument does not depend on the values of the indices \(i\), \(k\), we have

\[ \frac{A_{ik}}{A_{i+1\,k}} = \frac{A_{i\,k+1}}{A_{i+1\,k+1}} = \frac{A_{i\,k+2}}{A_{i+1\,k+2}} = \cdots = \frac{A_{it}}{A_{i+1\,t}}; \]

retaining only the first and the last ratios, we obtain

\[ \frac{A_{ik}}{A_{i+1\,k}} = \frac{A_{it}}{A_{i+1\,t}}. \]

In exactly the same way,

\[ \frac{A_{i+1\,k}}{A_{i+2\,k}} = \frac{A_{i+1\,t}}{A_{i+2\,t}}, \qquad \frac{A_{i+2\,k}}{A_{i+3\,k}} = \frac{A_{i+2\,t}}{A_{i+3\,t}}, \quad \ldots \ldots \ldots . \]

\[ \ldots \ldots \ldots \frac{A_{q-1\,k}}{A_{q\,k}} = \frac{A_{q-1\,t}}{A_{q\,t}}. \]

Multiplying these ratios, we find

\[ \frac{A_{ik}}{A_{q\,k}} = \frac{A_{it}}{A_{q\,t}}, \]

which was to be proved.

Theorem of Cayley—E. von Weber.

If all related determinants of order \((p+m)\), obtained by \(m\)-fold bordering of a determinant \(D\), nonzero and of order \(p\), are equal to zero, then all related determinants of order \((p+m)\) are also equal to zero*).

*) Ibid.

1. The combination of the Capelli–Garbieri and Cayley theorems, with additional arguments, gives a generalization of V. L. Nekrasov’s theorems*); we shall show this.

Take the matrix

Fig. 3.

\[ (\Phi)\qquad \left| \begin{array}{cccc|ccc} a_{11} & a_{12} & \cdots & a_{1,n-1} & a_{1n} & \cdots & a_{1m}\\ a_{21} & a_{22} & \cdots & a_{2,n-1} & a_{2n} & \cdots & a_{2m}\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots\\ a_{n-2,1} & a_{n-2,2} & \cdots & a_{n-2,n-1} & a_{n-2,n} & \cdots & a_{n-2,m}\\ a_{n-1,1} & a_{n-1,2} & \cdots & a_{n-1,n-1} & a_{n-1,n} & \cdots & a_{n-1,m}\\ a_{n1} & a_{n2} & \cdots & a_{n,n-1} & a_{nn} & \cdots & a_{nm} \end{array} \right| \]
\[ \underbrace{\hspace{7.2cm}}_{\Theta} \]

a) Let the determinant \(D=(a_{21}\ a_{32}\ \ldots\ a_{n,n-1})\ne 0\). If all the related determinants of order \(n\), obtained by bordering \(D\), are equal to \(0\), then, on the basis of Cayley’s theorem, all related determinants of order \(n\) are equal to \(0\).

For \(m=4\) and \(n=3\) we obtain the first theorem of V. L. Nekrasov.

b) The second theorem is of special interest, since it is not a consequence of any one of the general theorems.

Suppose that the results of a single bordering of \(D\) are all equal to zero, and at least one of the related determinants of order \(n\) is not equal to zero. This is possible only when \(D=0\), since otherwise, on the basis of Cayley’s theorem, all determinants of order \(n\) would be equal to zero.

It suffices to look at the matrix \((\Phi)\) in order to notice that every determinant obtained by a simple bordering of

\[ D=(a_{21}\ a_{32}\ \ldots\ a_{n,n-1}) \]

is at the same time the result of bordering the determinant

\[ D_{1}=(a_{11}\ a_{22}\ a_{n-1,n-1}); \]

therefore \(D_{1}=0\) as well, since otherwise all determinants of order \(n\) would be equal to zero.

Let us now direct our attention only to that part \((\Theta)\) of the matrix \((\Phi)\) which consists of the first \((n-1)\) columns, and which we have separated off by a line (see Fig. 3). We shall show that, when \(D=D_{1}=0\), all determinants of order \((n-1)\) of the matrix \((\Theta)\) are equal to zero.

Indeed, if

\[ \Delta=(a_{21}\ a_{23}\ldots a_{n-1,n-2})\ne 0, \]

then all determinants of order \((n-1)\) of the matrix are equal to zero by Cayley’s theorem: all,—in the present case \(D\) and \(D_{1}\),—determinants which are the result of bordering \(\Delta\), are equal to zero.

\[ \text{*) See the work named above.} \]

If \(\Delta=0\), then Cayley’s theorem cannot be applied, and additional arguments are required\(^*)\).

Let us denote by \(\Delta_1\) the determinant of order \(n\) contained in \((\Theta)\) and distinct from \(D\) and \(D_1\), and let us show that \(\Delta_1=0\). We note that \(\Delta_1\) must contain the 1st and \(n\)-th rows of the matrix \((\Theta)\), since otherwise it would coincide with \(D\) or \(D_1\), and also some \((n-3)\) rows among the \((n-1)\) rows: the 2nd, 3rd, etc., up to the \((n-1)\)-st.

Since \(D=0\), there exists one and the same linear relation among the elements of each of its columns. Therefore

\[ \begin{aligned} a_{11}&=\lambda_1 a_{21}+\lambda_2 a_{31}+\cdots+\lambda_{n-2}a_{n-1,1}\\ a_{12}&=\lambda_1 a_{22}+\lambda_2 a_{32}+\cdots+\lambda_{n-2}a_{n-1,2}\\ &\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\\ a_{1,n-1}&=\lambda_1 a_{2,n-1}+\lambda_2 a_{3,n-1}+\cdots+\lambda_{n-2}a_{n-1,n-1}; \end{aligned} \tag{1} \]

for \(D_1=0\) we have

\[ \begin{aligned} a_{n1}&=\mu_1 a_{21}+\mu_2 a_{31}+\cdots+\mu_{n-2}a_{n-1,1},\\ a_{n2}&=\mu_1 a_{22}+\mu_2 a_{32}+\cdots+\mu_{n-2}a_{n-1,2},\\ &\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots,\\ a_{n,n-1}&=\mu_1 a_{2,n-1}+\mu_2 a_{3,n-1}+\cdots+\mu_{n-2}a_{n-1,n-1}. \end{aligned} \tag{2} \]

Suppose, without loss of generality, that \(\Delta_1\) does not contain the second row of the matrix \((\Theta)\), i.e.

\[ \Delta_1= \begin{vmatrix} a_{11} & a_{12} & \cdots & a_{1,n-1}\\ a_{31} & a_{32} & \cdots & a_{3,n-1}\\ \ldots & \ldots & \ldots & \ldots\\ a_{n-1,1} & a_{n-1,2} & \cdots & a_{n-1,n-2}\\ a_{n1} & a_{n2} & \cdots & a_{n,n-1} \end{vmatrix}; \]

from the 1st row subtract the 2nd, 3rd, \(\ldots\), \((n-2)\)-nd, multiplied respectively by \(\lambda_2,\lambda_3,\ldots,\lambda_{n-2}\); from the \((n-1)\)-st row subtract the 2nd, 3rd, \(\ldots\), \((n-2)\)-nd, multiplied respectively by \(\mu_2,\mu_3,\ldots,\mu_{n-2}\); then, on the basis of equalities (1) and (2), we obtain

\[ \Delta_1= \begin{vmatrix} \lambda_1 a_{21} & \lambda_1 a_{22} & \cdots & \lambda_1 a_{2,n-1}\\ a_{31} & a_{32} & \cdots & a_{3,n-1}\\ \ldots & \ldots & \ldots & \ldots\\ a_{n-1,1} & a_{n-1,2} & \cdots & a_{n-1,n-2}\\ \mu_1 a_{21} & \mu_1 a_{22} & \cdots & \mu_1 a_{2,n-1} \end{vmatrix} =0, \]

since the elements of the 1st row are proportional to the elements of the last.

\(^*)\) In V. L. Nekrasov’s proof these arguments are concluded, for \(\Delta_2\ne0\), as a consequence of equalities (3) and (4); unfortunately, we could not apply V. L. Nekrasov’s method in the general case.

Thus, the determinants of order \((n-1)\) of the matrix \((\Theta)\) are all equal to \(0\) (in V. L. Nekrasov’s proof, what has just been set forth corresponds to the vanishing of the minors \((2)\)).

Consequently, the rank of the matrix \((\Theta)\) is \(r<n-1\), in particular equal to \((n-2)\); then—on the basis of the Cappelli–Garbieri theorem—all the associated determinants of its \(r\)-th order, contained in \(r\) rows (columns), are proportional to the corresponding determinants contained in any rows (columns) (to this last argument there corresponds the last equality in V. L. Nekrasov’s work).

Vl. Zylev.

15 January 1915.
Tomsk.