Reduction of a bilinear form to a canonical form

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D.V. Ionescu

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D.V. Ionescu
Institutul de Calcul

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D.V. Ionescu, Reducerea unei forme biliniare la o formă canonică. (Romanian) Stud. Cerc. Mat. (Cluj) 10 1959 45–49.

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Studii si Cercetari Matematice

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Academy of the Republic of S.R.

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https://ictp.acad.ro/scm/journal/article/view/42

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1220-269X

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MR0130264

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REDUCTION OF A BILINEAR FORM TO A CANONICAL FORM

OFDV IONESCU

In a note in the Bulletin of the Polish Academy of Sciences, from 1957, M. Altman [1] gave a generalization of Jacobi's method for bilinear forms. We take the liberty of pointing out that we presented at the first scientific session of the Society of Mathematical and Physical Sciences of the RPR, in 1955 [2], a paper containing as a particular case the reduction of a bilinear form to a canonical form. In this note we give another method for obtaining the canonical form.

Let us consider the bilinear form

\begin{matrix} (1) & F = \sum_{i, n = 1}^{n} a_{i, k} x_{i} {and}_{k} \end{matrix}

and either

\begin{matrix} (2) & X_{k} = \sum_{j = 1}^{n} a_{j, k} x_{j}, {AND}_{i} = \sum_{j = 1}^{n} a_{i, j} {and}_{j} \end{matrix}

where

i, k = 1, 2, \dots, n

.
Either

r

matrix rank

{‖ a_{i k} ‖}_{1}^{n}

and suppose that the notations were chosen so that the determinants

\begin{matrix} (3) & D_{h} = | \begin{array}{ccc} a_{11} & \dots & a_{1 h} \\ \cdot & \dots & \cdot \\ a_{h 1} & \dots & a_{h h} \end{array} | \end{matrix}

not be null, which is always possible, where

h = 1, 2, \dots, r

.

In this case it is demonstrated that

\begin{matrix} (4) & F = \frac{- 1}{D_{r}} | \begin{array}{cccc} a_{11} & \dots & a_{1 r} & {AND}_{1} \\ \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot \\ a_{r 1} & \dots & a_{r r} & {AND}_{r} \\ X_{1} & \dots & X_{r} & 0 \end{array} \end{matrix}

Applying Gauss' algorithm (see [3]) to solve the linear equations (2) in

{and}_{k}

and

x_{i}

, these equations can be written in the form

\begin{array}{r} a_{11} {and}_{1} + a_{12} {and}_{2} + \dots + a_{1 n} {and}_{n} = {AND}_{1} \\ a_{22}^{(1)} {and}_{2} + \dots + a_{2 n}^{(1)} {and}_{n} = {AND}_{2}^{(1)} \\ a_{n 2}^{(1)} {and}_{2} + \dots + a_{n n}^{(1)} {and}_{n} = {AND}_{n}^{(1)} \\ a_{11} x_{1} + a_{21} x_{2} + \dots + a_{n 1} x_{n} = X_{1} \\ a_{22}^{(1)} x_{2} + \dots + a_{n 2}^{(1)} x_{n} = X_{2}^{(1)} \\ a_{2 n}^{(1)} x_{2} + \dots + a_{n n}^{(1)} x_{n} = X_{n}^{(1)} \end{array}

and

a_{i k}^{(1)} = \frac{| \begin{matrix} a_{11} a_{1 k} \\ a_{i 1} a_{1 k} \end{matrix} |}{D_{1}}, X_{k}^{(1)} = \frac{| \begin{array}{cc} a_{11} a_{1 k} \\ X_{1} X_{k} \end{array} |}{D_{1}}, {AND}_{i}^{(1)} = \frac{| \begin{matrix} a_{11} {AND}_{1} \\ a_{i 1} {AND}_{i} \end{matrix} |}{D_{1}}

where

It is then demonstrated that

F = \frac{- 1}{D_{r}} | \begin{array}{ccccc} a_{11} & 0 & \dots & 0 & {AND}_{1} \\ 0 & a_{22}^{(1)} & \dots & a_{2 r}^{(1)} & {AND}_{2}^{(1)} \\ \cdot & \cdot & \dots & \cdot & \cdot \\ 0 & a_{r 2}^{(1)} & \dots & a_{r r}^{(1)} & {AND}_{r}^{(1)} \\ X_{1} & X_{2}^{(1)} & \dots & X_{r}^{(1)} & 0 \end{array} |

Continuing Gauss' algorithm, we arrive at the identity

F = \frac{- 1}{D_{r}} | \begin{array}{cccccc} a_{11} & 0 & 0 & \dots & 0 & {AND}_{1} \\ 0 & a_{22}^{(1)} & 0 & \dots & 0 & {AND}_{2}^{(1)} \\ 0 & 0 & a_{33}^{(2)} & \dots & 0 & {AND}_{3}^{(2)} \\ \dots & \dots & \dots & \dots & \dots & . \end{array} |

where

{AND}_{h}^{(h - 1)}

and

X_{h}^{(h - 1)}

are the second members of the linear equations

\begin{aligned} a_{11} {and}_{1} + a_{12} {and}_{2} + \dots + a_{1 n} {and}_{n} & = {AND}_{1} \\ a_{22}^{(1)} {and}_{2} + \dots + a_{2 n}^{(1)} {and}_{n} & = {AND}_{2}^{(1)} \\ a_{r r}^{(r - 1)} {and}_{r} + \dots + a_{r n}^{(r - 1)} {and}_{n} & = {AND}_{r}^{(r - 1)} \\ a_{11} x_{1} + a_{21} x_{2} + \dots + a_{n 1} x_{n} & = X_{1} \\ a_{22}^{(1)} x_{2} + \dots + a_{n 2}^{(1)} x_{n} & = X_{2}^{(1)} \\ a_{r r}^{(r - 1)} x_{r} + \dots + a_{n r}^{(r - 1)} x_{n} & = X_{r}^{(r - 1)} \end{aligned}

and
Gauss's.
obtained by Gauss's algorithm.
It is shown that

\begin{matrix} (6) & X_{k}^{(k - 1)} = \frac{P_{k} (x)}{Δ_{k}}, Y_{i}^{(i - 1)} = \frac{Q_{i} (x)}{Δ_{i}}, a_{h h}^{(h - 1)} = \frac{Δ_{h}}{Δ_{h - 1}} \end{matrix}

where

\begin{matrix} (7) & P_{k} (x) = | \begin{array}{ccc} a_{11} & \dots & a_{1 k} \\ \cdot & \dots & \cdot \\ a_{k - 1, 1} & \dots & a_{k - 1, k} \\ X_{1} & \dots & X_{k} \end{array} |, Q_{i} (y) = | \begin{array}{ccc} a_{11} & \dots & a_{1, i - 1} Y_{1} \\ \cdot & \cdot \\ \cdot & \cdot \\ a_{i 1} & \dots & a_{i, i - 1} Y_{i} \end{array} | \end{matrix}

Expanding the determinant from identity (5), we obtain the canonical form of the bilinear form (1)

Φ = \frac{X_{1} Y_{1}}{a_{11}} + \frac{X_{2}^{(1)} Y_{2}^{(1)}}{a_{22}^{(1)}} + \dots + \frac{X_{r}^{(r - 1)} Y_{r}^{(r - 1)}}{a_{r r}^{(r - 1)}}

which with the help of formulas (6) and (7) is written in the definitive form

\begin{matrix} (8) & Φ = \frac{P_{1} (x) Q_{1} (y)}{Δ_{0} Δ_{1}} + \frac{P_{2} (x) Q_{2} (y)}{Δ_{1} Δ_{2}} + \dots + \frac{P_{r} (x) Q_{r} (y)}{Δ_{r - 1} Δ_{r}} . \end{matrix}

It is easy to see that

P_{k} (x)

it does not depend on

x_{1}, x_{2}, \dots, x_{k - 1}

, that

Q_{i} (y)

it does not depend on

y_{1}, y_{2}, \dots, y_{i - 1}

and that

\frac{D (P_{1}, \dots, P_{r})}{D (x_{1}, \dots, x_{r})} = \frac{D (Q_{1}, \dots, Q_{r})}{D (y_{1}, \dots, y_{r})} = Δ_{1} Δ_{2} \dots Δ_{r} \neq 0,

which proves that linear forms

P_{k} (x)

and

Q_{i} (y)

are linearly independent.

When the bilinear form (1) is symmetric, that is,

a_{i k} = a_{k i}

, we have

P_{i} (x) = Q_{i} (x)

and formula (8) becomes

\begin{matrix} (9) & Φ = \frac{Q_{1} (x) Q_{1} (y)}{Δ_{0} Δ_{1}} + \frac{Q_{2} (x) Q_{2} (y)}{Δ_{1} Δ_{2}} + \dots + \frac{Q_{r} (x) Q_{r} (y)}{Δ_{r - 1} Δ_{r}} . \end{matrix}

In the particular case of quadratic shapes

x_{k} = y_{k}

, the previous formula gives Jacobi's classical formula for decomposing a quadratic form into a sum of squares

\begin{matrix} (10) & \sum_{i, k = 1}^{n} a_{i k} x_{i} x_{k} = \frac{Q_{1}^{2} (x)}{Δ_{0} Δ_{1}} + \frac{Q_{2}^{2} (x)}{Δ_{1} Δ_{2}} + \dots + \frac{Q_{r}^{2} (x)}{Δ_{r - 1} Δ_{r}} \end{matrix}

If the bilinear form (1) is with complex coefficients and indeterminates and in addition Hermitian, that is

a_{i k} = \overset{―}{a_{k i}}

, it can be written in the form

Φ_{1} = \sum_{i, k = 1}^{n} a_{i k} \overset{―}{x_{i}} y_{k} .

It is shown that in this case we have

P_{i} (x) = \overset{―}{Q_{i} (x)}

and the decomposition formula (8) becomes

Φ_{1} = \frac{\overset{―}{Q_{1}} (x) Q_{1} (y)}{Δ_{0} Δ_{1}} + \frac{\overset{―}{Q_{2} (x)} Q_{2} (y)}{Δ_{1} Δ_{2}} + \dots + \frac{\overset{―}{Q_{r} (x)} Q_{r} (y)}{Δ_{r - 1} Δ_{r}}

determinants

Δ_{i}

being real.
When

x_{i} = y_{i}

, the previous formula gives the decomposition of a Hermitian quadratic form into a sum of squares

\begin{matrix} (11) & \sum_{i, k = 1}^{n} a_{l k} {\bar{x}}_{i} x_{k} = \frac{{| Q_{1} (x) |}^{2}}{Δ_{0} Δ_{1}} + \frac{{| Q_{2} (x) |}^{2}}{Δ_{1} Δ_{2}} + \dots + \frac{{| Q_{r} (x) |}^{2}}{Δ_{r - 1} Δ_{r}} . \end{matrix}

BIBLIOGRAPHIC BOOKS

A1tman M., A generalisation of Jacobi's method for bilinear forms. Bull. de l'Acad. Polon. des Sci., 1957, tom. V, p. 99-104.
Ionescu DV, An important identity and the decomposition of a bilinear form into a sum of products. Gazeta matematica și fizica, Series A, 1955, pp. 303-312 (referred to in Reterativnîi Jurnal - Matematika, 1957, ref. no. 1146).
Gantmacher FR, Matrix Theory, Moscow, 1953, vol. II, p. 28 (lithographic transl. from the 1st Russian).

REDUCTION OF BILINEAR FORM TO CANONICAL FORM (Summary)

In a note from the Bulletin of the Polish Academy of Sciences for 1957, M. Altman [1] gave a generalization of Jacoba's method for bilinear forms. At the first scientific session of the PHP Society for Mathematical and Physical Sciences in 1955 [2], a paper was presented containing a special case of reducing a bilinear form to a conical form. This note presents a new method for obtaining the canonical form.

First, identity (5) is proved for the bilinear form (1), which implies the canonical form (8), where

P (x), Q (x)

And

Δ_{i}

are given by formulas (7) and (3). As a special case, we obtain formula (10), which represents the classical Jacobi decomposition for a quadratic form, as well as formula (11), which represents the decomposition of a Hermitian quadratic form into a sum of squares.

REDUCTION FROM A BILINERIC FORM TO A CANONICAL FORM
(Summary)

In a note in the Bulletin of the Polish Academy of Sciences, dated 1957, Mr. Altman [1] gave a generalization of Jacobi's method for bilinear forms. I would like to point out that I presented a paper at the first scientific session of the Society of Mathematical and Physical Sciences of the Polish People's Republic (RPR) in 1955 [2], which includes as a special case the reduction of a bilinear form to a canonical form, and in this note I give a summary of the method I used.

We first demonstrate, for the bilinear form (1), the identity (5), from which the canonical form (8) follows, the

P_{i} (x), Q_{i} (y)

And

Δ_{i}

given formulas (7) and (3). We obtain as a special case formula (10) which is the classical Jacobi decomposition of the quadratic form and formula (11) which is the decomposition of a Hermitian quadratic form into a sum of squares.

1959

Reduction of a bilinear form to a canonical form

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REDUCTION OF A BILINEAR FORM TO A CANONICAL FORM

BIBLIOGRAPHIC BOOKS

REDUCTION OF BILINEAR FORM TO CANONICAL FORM (Summary)

REDUCTION FROM A BILINERIC FORM TO A CANONICAL FORM
(Summary)

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Reduction of a bilinear form to a canonical form

Abstract

Authors

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Paper coordinates

PDF

About this paper

Journal

Publisher Name

DOI

Print ISSN

Online ISSN

References

Paper (preprint) in HTML form

REDUCTION OF A BILINEAR FORM TO A CANONICAL FORM

BIBLIOGRAPHIC BOOKS

REDUCTION OF BILINEAR FORM TO CANONICAL FORM (Summary)

REDUCTION FROM A BILINERIC FORM TO A CANONICAL FORM (Summary)

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REDUCTION FROM A BILINERIC FORM TO A CANONICAL FORM
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