- #1

- 240

- 42

$$|\det(A^tA)|=\sum_{1\le j_1\le ... \le j_n \le m} (det(A_{j_1...j_n}))^2$$

where ##A_{j_1...j_n}## is the matrix whose ##(i,k)##-entry is ##a_{j,k},## and ##A^t## is the transpose of ##A.##

The example of this is given here to make clear:

$$\det (

\begin{pmatrix}

1 & 3 &5\\

2 & 4 &6\\

\end{pmatrix}

\begin{pmatrix}

1 & 2 \\

3 & 4 \\

5 & 6

\end{pmatrix}) =

(\det\begin{pmatrix}

1 & 2 \\

3 & 4 \\

\end{pmatrix})^2+

(\det\begin{pmatrix}

1 & 2 \\

5 & 6 \\

\end{pmatrix})^2+

(\det\begin{pmatrix}

3 & 4 \\

5 & 6 \\

\end{pmatrix})^2

.$$

The equation is clearly holds fro square matrix, but for general type I cannot solve...I try to prove it in induction from ##m\times n## to ##(m+1)\times n## but failed. This may be related to the notion of area (the given example is the area of a triangle on a plane).

Thanks for any ideas in advance!!!