Determinant
Intuitive understanding of determinant
- Area(Volume) of a matrix
- How much a matrix modifies the area(volume) of a target matrix through linear transformation
The first meaning and the second are basically the same if you set identity matrix as the target matrix.
If \(det(A) > 0\), it changes the area(volume) of target matrix, preserving it’s direction.
If \(det(A) < 0\), it inverts the order of target matrix’s column vectors through linear transformation.
If \(det(A) = 0\), it squeezes some vectors to zero.
For example, Let \(A=\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\)
Then \(A\) transforms \(2\)-dimensional space into a line \(y=0\). In this case, \(det(A) = 0\)
Characteristics of determinant
-
commutative :
\(det(AB) = det(A)det(B) \: where \: A, B \in \mathbb{R}^{n\times n}\) -
cofactor :
\(det(A) = \sum_{j} a_{ij}A_{ij} \: where \: A_{ij} = (-1)^{i+j}det(M_{ij})\) -
decomposition :
\(det(A) = det(P^{-1}LU) = \pm (products \: of \: pivots)\) - permutation :
\(det(A) = \sum_{\sigma}a_{1\sigma_{1}}a_{2\sigma_{2}}\cdots a_{n\sigma_{n}}det(P_{\sigma}) \\ det(A) = a_{11}a_{22} - a_{12}a_{21}\) comes from this rule. -
if \(A\) is inversible :
\(A^{-1} = \dfrac{adj(A)}{det(A)}\) -
cramer’s rule :
\(x_j = \dfrac{det(B_j)}{det(A)} \\ where \: B_j = \begin{bmatrix} a_{11} & \cdots & b_1 & \cdots & a_{1n} \\ a_{21} & \cdots & b_2 & \cdots & a_{2n} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ a_{n1} & \cdots & b_n & \cdots & a_{nn} \end{bmatrix}\) -
Sylvester’s determinant identity :
\(det(I_m + AB^T) = det(I_n + B^TA) \\ where A,B \in \mathbb{R^{m \times n}}\) - Schur complement :
- Sherman-Woodbury-Morfison identity :
\((A + BD^{-1}C)^{-1} = A^{-1} -A^{-1}B(D + CA^{-1}B)^{-1}CA^{-1}\)
