Now \(N\times N\) matrices \(\tilde A=\left(a_{jk}\right)\;j,k=1,\ldots,N\)
Definition 25 \begin{eqnarray*} \tilde A&=&\left(\begin{array}{cccc}a_{11}&a_{12}&\cdots&a_{1N}\\a_{21}&a_{22}&\cdots&a_{2N}\\\vdots&\vdots&\ddots&\vdots\\a_{N1}&a_{N2}&\cdots&a_{NN}\end{array}\right)\\ \\ \det{\left(\tilde A\right)}&=&\left|\begin{array}{ccc}a_{11}&\cdots&a_{1N}\\\vdots&\ddots&\vdots\\a_{N1}&\cdots&a_{NN}\end{array}\right|=\sum_{P(N)}\left(-1\right)^{j\left(P\right)}a_{1,j_1}a_{2,j_2}\cdots a_{N,j_N}\\\\ & & \mbox{where P(N) are all permutations of the numbers $1,\ldots,N$}\\&&\mbox{and j(P) is the number of changes between $(1,\ldots,N)$ and $(j_1,\ldots,j_N)$ }\end{eqnarray*}
\(\Rightarrow\) definition not practical for a calculation of \(\det(\tilde A)\). Therefore, calculation via successive expansion in sub-determinants (Laplace rule): \(N=1:\; \det(a)=a \quad a\in\mathbb{R}\).
As we will see the determinant is the (only) totally antisymmetric multilinear operation acting on the components of a matrix. Corresponding to the Kronecker-symbol sometimes a notation using the totally antisymmetric function \(\epsilon_{i,j,...,k}\) (Levi-Civita symbol) is helpful for the formal calculation of a determinant:
| \[\det{\left(\tilde A\right)} = \sum_{P(N)}\left(-1\right)^{j\left(P\right)}a_{1,j_1}a_{2,j_2}\cdots a_{N,j_N} = \sum_{j_1, j_2, ..., j_N = 1}^N \epsilon_{j_1,j_2,...,j_N} a_{1,j_1}a_{2,j_2}\cdots a_{N,j_N}\] |
\(\epsilon_{j_1,j_2,...,j_N}\) is zero if any of the indices are equal, it
is \(1 = \epsilon_{1,2,...,N}\), and changes it’s sign for each change in the order of indices. (Hint:
this are exactly the properties of the quantum numbers of Fermions according to the Pauli principle, the determinant or
\(\epsilon_{i,j,...,k}\) are therefor often used to calculate many particle states in quantum mechanics).
The geometrical interpretation in 3D of a determinant will be given in section 2.15 and a more general interpretation in section 2.16.
Calculation by Laplace rule:
| \[\det\left(\tilde A\right)=\sum_{j=1}^N a_{jk}A_{jk}=\sum_{j=1}^N a_{kj}A_{kj},\mbox{ for }N\gt1\] |
development via the column/line, adaptive expansion by column or row where: \(k\in(1,\ldots,N)\)
arbitrary and
cofactor of \(a_{jk}\) in \(\tilde
A\)
| \[A_{jk}=\left(-1\right)^{j+k}\left|\begin{array}{ccccccc}a_{11}&a_{12}&\cdots&a_{1,k-1}&a_{1,k+1}&\cdots&a_{1N}\\ a_{21}&a_{22}&\cdots&a_{2,k-1}&a_{2,k+1}&\cdots&a_{2N}\\ \vdots\\ a_{j-1,1}&a_{j-1,2}&\cdots&a_{j-1,k-1}&a_{j-1,k+1}&\cdots&a_{j-1,N}\\ a_{j+1,1}&a_{j+1,2}&\cdots&a_{j+1,k-1}&a_{j+1,k+1}&\cdots&a_{j+1,N}\\ \vdots\\ a_{N1}&a_{N2}&\cdots&a_{N,k-1}&a_{N,k+1}&\cdots&a_{NN} \end{array}\right|\hat=\begin{array}{l}\mbox{determinants of $\tilde A$ where}\\\mbox{j$^{th}$ line and k$^{th}$column are}\\\mbox{erased}\end{array}\] |
Examples:
| \[\left|\begin{array}{cc}a_{11}&a_{12}\\a_{21}&a_{22}\end{array}\right|=a_{11}a_{22}-a_{21}a_{12}\quad\text{development via 1$^{\mbox{st}}$ column}\] |
\begin{eqnarray*}\left|\begin{array}{ccc} a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{array}\right|&=&a_{11}\left| {\genfrac{}{}{0pt}{}{a_{22}}{a_{32}}}{\genfrac{}{}{0pt}{}{a_{23}}{a_{33}}}\right|-a_{12}\left|{\genfrac{}{}{0pt}{}{a_{21}}{a_{31}}}{\genfrac{}{}{0pt}{}{a_{23}}{a_{33}}}\right|+a_{13}\left|{\genfrac{}{}{0pt}{}{a_{21}}{a_{31}}}{\genfrac{}{}{0pt}{}{a_{22}}{a_{32}}}\right|\\ &=&a_{11}a_{22}a_{33}-a_{11}a_{32}a_{23}-a_{12}a_{21}a_{33}+a_{12}a_{23}a_{31}+a_{12}a_{21}a_{32}-a_{13}a_{31}a_{22}\end{eqnarray*}
\(\Rightarrow\) calculation of larger determinants still difficult!
Calculation rules for determinants:
Determinants are antisymmetric for changing the order of rows or columns,
i.e. \(\quad \left|\begin{array}{ccc}\vec{a}&\vec{b}&\cdots\end{array}\right|=-\left|\begin{array}{ccc}\vec{b}&\vec{a}&\cdots\end{array}\right|\)
Determinants vanish if two vectors are identical,
i.e. \(\quad
\left|\begin{array}{ccc}\vec{a}&\vec{a}&\cdots\end{array}\right|=-\left|\begin{array}{ccc}\vec{a}&\vec{a}&\cdots\end{array}\right|=0\)
Determinants are linear,
i.e. \(\quad \left|\begin{array}{cc}\vec{a}+\vec{b}&\cdots\end{array}\right|=\left|\begin{array}{cc}\vec{a}&\cdots\end{array}\right|+\left|\begin{array}{cc}\vec{b}&\cdots\end{array}\right|\)
Determinants are linear,
i.e. \(\quad \left|\begin{array}{cc}\alpha
\vec{a}&\cdots\end{array}\right|=\alpha \left|\begin{array}{cc}\vec{a}&\cdots\end{array}\right|\)
Adding linear combination of other rows/columns does not change Determinants,
i.e. \(\quad \left|\begin{array}{ccc}\left(\vec{a}+\beta \vec{b}\right)&\vec{b}&\cdots\end{array}\right|=\left|\begin{array}{ccc}\vec{a}&\vec{b}&\cdots\end{array}\right|+\beta
\left|\begin{array}{ccc}\vec{b}&\vec{b}&\cdots\end{array}\right|=\left|\begin{array}{ccc}\vec{a}&\vec{b}&\cdots\end{array}\right|\)
Subtracting projections of (row/column)-vectors does therefor not change the determinant, so the determinant calculates the product of the length of a set of orthogonal vectors, i.e the volume spanned up by the set of vectors. If the volume is not zero the set of vectors is linearly independent.
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© J. Carstensen (Math for MS)