Cauchy-Binet Formula/Example

Examples of Cauchy-Binet Formula

Cauchy-Binet Formula: $m = n$

Let $\mathbf A = \sqbrk a_n$ and $\mathbf B = \sqbrk b_n$ be a square matrices of order $n$.

Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$.

Let $\mathbf A \mathbf B$ be the (conventional) matrix product of $\mathbf A$ and $\mathbf B$.

Then:

$\map \det {\mathbf A \mathbf B} = \map \det {\mathbf A} \map \det {\mathbf B}$

That is, the determinant of the product is equal to the product of the determinants.

Cauchy-Binet Formula: $m = 1$

Let $\mathbf A = \sqbrk a_{1 n}$ be a row matrix with $n$ columns.

and $\mathbf B = \sqbrk b_{n 1}$ be a column matrix with $n$ rows.

Let $\mathbf A \mathbf B$ be the (conventional) matrix product of $\mathbf A$ and $\mathbf B$.

Then:

$\ds \map \det {\mathbf A \mathbf B} = \sum_{j \mathop = 1}^n a_j b_j$

where:

$a_j$ is element $a_{1 j}$ of $\mathbf A$

$b_j$ is element $b_{j 1}$ of $\mathbf B$.

Cauchy-Binet Formula: Matrix by Transpose

Let $\mathbf A$ be an $m \times n$ matrix.

Let $\mathbf A^\intercal$ be the transpose $\mathbf A$.

Let $1 \le j_1, j_2, \ldots, j_m \le n$.

Let $\mathbf A_{j_1 j_2 \ldots j_m}$ denote the $m \times m$ matrix consisting of columns $j_1, j_2, \ldots, j_m$ of $\mathbf A$.

Let $\mathbf A^\intercal_{j_1 j_2 \ldots j_m}$ denote the $m \times m$ matrix consisting of rows $j_1, j_2, \ldots, j_m$ of $\mathbf A^\intercal$.

Then:

$\ds \map \det {\mathbf A \mathbf A^\intercal} = \sum_{1 \mathop \le j_1 \mathop < j_2 \mathop < \cdots \mathop < j_m \le n} \paren {\map \det {\mathbf A_{j_1 j_2 \ldots j_m} } }^2$

where $\det$ denotes the determinant.

Cauchy-Binet Formula: $m > n$

Let $\mathbf A$ be an $m \times n$ matrix.

Let $\mathbf B$ be an $n \times m$ matrix.

Let $m > n$.

Then:

$\map \det {\mathbf A \mathbf B} = 0$

Cauchy-Binet Formula: $m = 2$

$\ds \paren {\sum_{i \mathop = 1}^n a_i c_i} \paren {\sum_{j \mathop = 1}^n b_j d_j} = \paren {\sum_{i \mathop = 1}^n a_i d_i} \paren {\sum_{j \mathop = 1}^n b_j c_j} + \sum_{1 \mathop \le i \mathop < j \mathop \le n} \paren {a_i b_j - a_j b_i} \paren {c_i d_j - c_j d_i}$

where all of the $a, b, c, d$ are elements of a commutative ring.

Thus the identity holds for $\Z, \Q, \R, \C$.