Determinant

  • For an n×nn \times n matrix AA, the determinant of AA denotes the magnification of the volume of nn-dimensional parallelopiped.
  • More formally, the following are the determinants of 2×22 \times 2 and 3×33 \times 3 matrices.

Ratio

If A=(a1,a2)A = (a_1, a_2) is a 2×22 \times 2 matrix, detA\det A means the transformed area from the unit area. If A=(a1,a2,a3)A = (a_1, a_2, a_3) is a 3×33 \times 3 matrix, detA\det A means the transformed volume form the unit volume.

  • For the arbitrary column vector aia_i of AA, adding kaika_i to another column vector aja_j does not change detA\det A. For example, det(a1,a2,a3)=det(a1,a2,a3+ca2)\det (a_1, a_2, a_3) = \det (a_1, a_2, a_3 + ca_2) for cRc \in \R. Consider detA\det A as a deck, then its volume is not changed although it is pushed to some directions.

Volume

  • The determinant of an upper triangular matrix AA is the product of all the diagonal elements of AA.
A=(a11a12a130a22a2300a33)    detA=a11a22a33\begin{aligned} A &= \left(\begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ 0 & a_{22} & a_{23} \\ 0 & 0 & a_{33} \end{array}\right) \implies \det A = a_{11}a_{22}a_{33} \end{aligned}

UpperTriangular

  • For A=(a1,,an)A = (a_1, \cdots, a_n) and cRc \in \R, det(ca1,a2,,an)=cdet(a1,,an)\det (ca_1, a_2, \cdots, a_n) = c \det (a_1, \cdots, a_n). Moreover, det(a1+a1,a2,,an)=det(a1,a2,,an)+det(a1,a2,,an)\det (a_1 + a'_1, a_2, \cdots, a_n) = \det (a_1, a_2, \cdots, a_n) + \det (a'_1, a_2, \cdots, a_n) for an arbitrary n×1n \times 1 vector a1a'_1.

Deformation

Reference

[1] Hiraoka Kazuyuki, Hori Gen, Programming No Tame No Senkei Daisu, Ohmsha.


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