صيغة كوشي-بينيه

في الجبر الخطي، صيغة كوشي-بينيه Cauchy–Binet formula هي الصيغة التي تعمم قاعدة جداء المحددات (وهي التي تقول أن محدد ناتج جداء مصفوفتين مربعتين يساوي إلى جداء محدديهما) لتطبق على مصفوفات غير مربعة .

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 العبارة

لنفرض أن A مصفوفة m×n و B مصفوفة n×m . اذا كان S مجموعة جزئية من { 1, ..., n } ذات m عنصر, يمكننا أن نكتب A_S من أجل المصفوفة m×m التي أعمدتها هي الأعمدة A ذات الأدلة من S. بشكل مشابه ، يمكن ان نكتب أن B_S من أجل المصفوفة m×m التي صفوفها هي صفوف B ذات الأدلة من S. تقول عندها صيغة كوشي-بينيه:

$\det(AB)=\sum _{S}\det(A_{S})\det(B_{S})\,$

حيث المجموع يمدد على كل المجموعات الجزئية S من { 1, ..., n } ذات m عنصر (هناك C(n,m) لجميع ما ذكرنا).

Example: Taking m = 2 and n = 3, and matrices $A={\begin{pmatrix}1&1&2\\3&1&-1\\\end{pmatrix}}$ and $B={\begin{pmatrix}1&1\\3&1\\0&2\end{pmatrix}}$، صيغة كوشي-بينيه تعطي المحددة

$\det(AB)=\left|{\begin{matrix}1&1\\3&1\end{matrix}}\right|\cdot \left|{\begin{matrix}1&1\\3&1\end{matrix}}\right|+\left|{\begin{matrix}1&2\\1&-1\end{matrix}}\right|\cdot \left|{\begin{matrix}3&1\\0&2\end{matrix}}\right|+\left|{\begin{matrix}1&2\\3&-1\end{matrix}}\right|\cdot \left|{\begin{matrix}1&1\\0&2\end{matrix}}\right|.$

Indeed $AB={\begin{pmatrix}4&6\\6&2\end{pmatrix}}$ ، ومحددتها هي $-28$ التي تساوي $-2\times -2+-3\times 6+-7\times 2$ من الجانب الأيمن للصيغة.

 حالات خاصة

If n < m then ${\tbinom {[n]}{m}}$ is the empty set, and the formula says that det(AB) = 0 (its right hand side is an empty sum); indeed in this case the rank of the m×m matrix AB is at most n, which implies that its determinant is zero. If n = m, the case where A and B are square matrices, ${\tbinom {[n]}{m}}=\{[n]\}$ (a singleton set), so the sum only involves S = [n], and the formula states that det(AB) = det(A)det(B).

For m = 0, A and B are empty matrices (but of different shapes if n > 0), as is their product AB; the summation involves a single term S = Ø, and the formula states 1 = 1, with both sides given by the determinant of the 0×0 matrix. For m = 1, the summation ranges over the collection ${\tbinom {[n]}{1}}$ of the n different singletons taken from [n], and both sides of the formula give $\textstyle \sum _{j=1}^{n}A_{1,j}B_{j,1}$, the dot product of the pair of vectors represented by the matrices. The smallest value of m for which the formula states a non-trivial equality is m = 2; it is discussed in the article on the Binet–Cauchy identity.

 في حالة n = 3

Let ${\boldsymbol {a}},{\boldsymbol {b}},{\boldsymbol {c}},{\boldsymbol {d}},{\boldsymbol {x}},{\boldsymbol {y}},{\boldsymbol {z}},{\boldsymbol {w}}$ be three-dimensional vectors.

${\begin{aligned}&1=1&(m=0)\\[10pt]&{\boldsymbol {a}}\cdot {\boldsymbol {x}}=a_{1}x_{1}+a_{2}x_{2}+a_{3}x_{3}&(m=1)\\[10pt]&{\begin{vmatrix}{\boldsymbol {a}}\cdot {\boldsymbol {x}}&{\boldsymbol {a}}\cdot {\boldsymbol {y}}\\{\boldsymbol {b}}\cdot {\boldsymbol {x}}&{\boldsymbol {b}}\cdot {\boldsymbol {y}}\end{vmatrix}}\\[4pt]={}&{\begin{vmatrix}a_{2}&a_{3}\\b_{2}&b_{3}\end{vmatrix}}{\begin{vmatrix}x_{2}&y_{2}\\x_{3}&y_{3}\end{vmatrix}}+{\begin{vmatrix}a_{3}&a_{1}\\b_{3}&b_{1}\end{vmatrix}}{\begin{vmatrix}x_{3}&y_{3}\\x_{1}&y_{1}\end{vmatrix}}+{\begin{vmatrix}a_{1}&a_{2}\\b_{1}&b_{2}\end{vmatrix}}{\begin{vmatrix}x_{1}&y_{1}\\x_{2}&y_{2}\end{vmatrix}}\\[4pt]={}&({\boldsymbol {a}}\times {\boldsymbol {b}})\cdot ({\boldsymbol {x}}\times {\boldsymbol {y}})&(m=2)\\[10pt]&{\begin{vmatrix}{\boldsymbol {a}}\cdot {\boldsymbol {x}}&{\boldsymbol {a}}\cdot {\boldsymbol {y}}&{\boldsymbol {a}}\cdot {\boldsymbol {z}}\\{\boldsymbol {b}}\cdot {\boldsymbol {x}}&{\boldsymbol {b}}\cdot {\boldsymbol {y}}&{\boldsymbol {b}}\cdot {\boldsymbol {z}}\\{\boldsymbol {c}}\cdot {\boldsymbol {x}}&{\boldsymbol {c}}\cdot {\boldsymbol {y}}&{\boldsymbol {c}}\cdot {\boldsymbol {z}}\end{vmatrix}}={\begin{vmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\end{vmatrix}}{\begin{vmatrix}x_{1}&y_{1}&z_{1}\\x_{2}&y_{2}&z_{2}\\x_{3}&y_{3}&z_{3}\end{vmatrix}}\\[4pt]={}&[{\boldsymbol {a}}\cdot ({\boldsymbol {b}}\times {\boldsymbol {c}})][{\boldsymbol {x}}\cdot ({\boldsymbol {y}}\times {\boldsymbol {z}})]&(m=3)\\[10pt]&{\begin{vmatrix}{\boldsymbol {a}}\cdot {\boldsymbol {x}}&{\boldsymbol {a}}\cdot {\boldsymbol {y}}&{\boldsymbol {a}}\cdot {\boldsymbol {z}}&{\boldsymbol {a}}\cdot {\boldsymbol {w}}\\{\boldsymbol {b}}\cdot {\boldsymbol {x}}&{\boldsymbol {b}}\cdot {\boldsymbol {y}}&{\boldsymbol {b}}\cdot {\boldsymbol {z}}&{\boldsymbol {b}}\cdot {\boldsymbol {w}}\\{\boldsymbol {c}}\cdot {\boldsymbol {x}}&{\boldsymbol {c}}\cdot {\boldsymbol {y}}&{\boldsymbol {c}}\cdot {\boldsymbol {z}}&{\boldsymbol {c}}\cdot {\boldsymbol {w}}\\{\boldsymbol {d}}\cdot {\boldsymbol {x}}&{\boldsymbol {d}}\cdot {\boldsymbol {y}}&{\boldsymbol {d}}\cdot {\boldsymbol {z}}&{\boldsymbol {d}}\cdot {\boldsymbol {w}}\end{vmatrix}}=0&(m=4)\end{aligned}}$

في حالة m > 3 ، الجانب الأيمن دائماً يساوي 0.

 برهان بسيط

The following simple proof presented in relies on two facts that can be proven in several different ways:

1. For any $1\leq k\leq n$ the coefficient of $z^{n-k}$ in the polynomial $\det(zI_{n}+X)$ is the sum of the $k\times k$ principal minors of $X$.
2. If $m\leq n$ and $A$ is an $m\times n$ matrix and $B$ an $n\times m$ matrix, then

$\det(zI_{n}+BA)=z^{n-m}\det(zI_{m}+AB)$.

Now, if we compare the coefficient of $z^{n-m}$ in the equation $\det(zI_{n}+BA)=z^{n-m}\det(zI_{m}+AB)$, the left hand side will give the sum of the principal minors of $BA$ while the right hand side will give the constant term of $\det(zI_{m}+AB)$, which is simply $\det(AB)$, which is what the Cauchy–Binet formula states, i.e.

${\begin{aligned}&\det(AB)=\sum _{S\in {\tbinom {[n]}{m}}}\det((BA)_{S,S})=\sum _{S\in {\tbinom {[n]}{m}}}\det(B_{S,[m]})\det(A_{[m],S})\\[5pt]={}&\sum _{S\in {\tbinom {[n]}{m}}}\det(A_{[m],S})\det(B_{S,[m]}).\end{aligned}}$

 العلاقة مع دلتا كرونيكر المعممة

As we have seen, the Cauchy–Binet formula is equivalent to the following:

$\det(L_{f}R_{g})=\sum _{S\in {\tbinom {[n]}{m}}}\det((L_{f})_{[m],S})\det((R_{g})_{S,[m]}),$

حيث

$L_{f}={\bigl (}(\delta _{f(i),j})_{i\in [m],j\in [n]}{\bigr )}\quad {\text{and}}\quad R_{g}={\bigl (}(\delta _{j,g(k)})_{j\in [n],k\in [m]}{\bigr )}.$

In terms of generalized Kronecker delta, we can derive the formula equivalent to the Cauchy–Binet formula:

$\delta _{g(1)\dots g(m)}^{f(1)\dots f(m)}=\sum _{k:[m]\to [n] \atop k(1)<\dots <k(m)}\delta _{k(1)\dots k(m)}^{f(1)\dots f(m)}\delta _{g(1)\dots g(m)}^{k(1)\dots k(m)}.$

 تفسيرات هندسية

If A is a real m×n matrix, then det(A A^T) is equal to the square of the m-dimensional volume of the parallelotope spanned in Rⁿ by the m rows of A. Binet's formula states that this is equal to the sum of the squares of the volumes that arise if the parallelepiped is orthogonally projected onto the m-dimensional coordinate planes (of which there are ${\tbinom {n}{m}}$).

In the case m = 1 the parallelotope is reduced to a single vector and its volume is its length. The above statement then states that the square of the length of a vector is the sum of the squares of its coordinates; this is indeed the case by the definition of that length, which is based on the Pythagorean theorem.

 التعميم

The Cauchy–Binet formula can be extended in a straightforward way to a general formula for the minors of the product of two matrices. Context for the formula is given in the article on minors, but the idea is that both the formula for ordinary matrix multiplication and the Cauchy-Binet formula for the determinant of the product of two matrices are special cases of the following general statement about the minors of a product of two matrices. Suppose that A is an m × n matrix, B is an n × p matrix, I is a subset of {1,...,m} with k elements and J is a subset of {1,...,p} with k elements. Then

$[\mathbf {AB} ]_{I,J}=\sum _{K}[\mathbf {A} ]_{I,K}[\mathbf {B} ]_{K,J}\,$

where the sum extends over all subsets K of {1,...,n} with k elements.

 الهامش

• Joel G. Broida & S. Gill Williamson (1989) A Comprehensive Introduction to Linear Algebra, §4.6 Cauchy-Binet theorem, pp 208–14, Addison-Wesley ISBN 0-201-50065-5.
• Jin Ho Kwak & Sungpyo Hong (2004) Linear Algebra 2nd edition, Example 2.15 Binet-Cauchy formula, pp 66,7, Birkhäuser ISBN 0-8176-4294-3.
• I. R. Shafarevich & A. O. Remizov (2012) Linear Algebra and Geometry, §2.9 (p. 68) & §10.5 (p. 377), Springer ISBN 978-3-642-30993-9.

 وصلات خارجية

• Aaron Lauve (2004) A short combinatoric proof of Cauchy–Binet formula from Université du Québec à Montréal.