Determinats(行列式) 2018-11-23

作者: 默写年华Antifragile | 来源:发表于2018-11-27 11:15 被阅读15次

1. Main Use of Determinants

They test for invertibility. If the determinants of A is zero, then A is singular. If detA ≠0, then A is invertible.
The determinant of A equals the volume of a box in n-dimensional space. The edges of the box come from the rows of A. The columns of A would give an entirely different box with the same volume.
The determinants gives a formula for each pivots.
The determinant measures the dependence of $A^{-1}b$ on each element of b. If one parameter is changed in an experiment, or one observation is corrected, the "influence coefficient" in $A^{-1}$ is a ratio of determinants.

2. Properties of the Determinant

The determinant of the identity matrix is 1
The determinant changes sign when two rows are exchanged.
The determinant of every permutation matrix is det P=±1. By row exchanges, we can turn P into the identity matrix.
The determinant is linear in each row separately
If two rows of A are equal, then detA =0
Subtracting a multiple of one row from another row leaves the same determinant. (The usual elimination steps do not affect the determinant)
If A has a row of zeros, then det A = 0
If A is triangular then det A is the product $a_{11}a_{22}a_{33}...a_{nn}$ of diagonal entries. If triangular A has 1s along the diagonal, then det A = 1
If A is singular, then det A = 0. If A is invertible , then det A ≠ 0.
The determinant of AB is the product of det A and det B
product rule: |A||B|=|AB|
The transpose of A has the same determinant as A itself: $detA^T=det A$
From this point, every rule that applied to the rows can now be applied to the columns: The determinant change sign when two columns are exchanged, two equal columns (or a column of zeros) produce a zero determinant, and the determinant depends linearly on each individual column

3. Formulas for the Determinants

If A is invertible, then PA=LDU and det P=+1. The product rule gives $det A=±det L det D det U=±$ (productof pivots)
The sign ±1 depends on whether the number of row exchanges is even or odd. The triangular factors have det L=det U =1 and det D=d1...dn
The determinant of A is a combination of any row i times its cofactors:
det A by cofactors: $det A=a_{i1}C_{i1}+a_{i2}C_{i2}+....+a_{in}C_{in}$
The cofactor $C_{ij}$ is the determinant of $M_{ij}$ with the correct sign:
delete row i and column j $C_{ij}=(-1)^{i+j}detM_{ij}$
These formulas express detA as a combination of determinants of order n-1

4. Applications of Determinants

4.1 Computation of $A^{-1}$

Cofactor matrix, C is transposed
$A^{-1}=\frac{C^T}{detA}$ means $A^{-1}_{ij}=\frac{C_{ji}}{detA}$

4.2 The solution of Ax=b: Cramer's rule

The jth component of $x= A^{-1}b$ is the ratio
$x_j=\frac{det B_j}{detA}$ where (has b in column j) $B_j= \left[ \begin{matrix} a_{11}&a_{12}&b_1&a_{1n}\\ a_{21}&a_{22}&b_2&a_{2n} \\ \vdots & \vdots & \vdots & \vdots\\ a_{n1}&a_{n2}&b_n&a_{nn} \end{matrix} \right]$

4.3 The Volume of a Box

The determinant equals the volume

4.4 A Formula for the Pivots

If A is factored into LDU, the upper left corners satisfy $A_k=L_kD_KU_k$ . For every k, the submatrix $A_k$ is going through a Gaussian elimination of its own.
Formula for pivots: $\frac{detA_k}{detA_{k-1}}=\frac{d_1d_2\cdots d_k}{d_1d_2\cdots d_{k-1}}=d_k$ (By convention, $detA_0=1$ )
Mutiplying together all the individual pivots, we recover:
$d_1d_2\cdots d_n=\frac{detA_1}{detA_0}\frac{detA_2}{detA_1}\cdots\frac{detA_n}{detA_{n-1}}=\frac{detA_n}{detA_0}=det A$
The pivot entries are all nonzero whenever the number $detA_k$ are all nonzero
Elimination can be completed without row exchanges (so P=I and A = LU), if and only if the leading submatrices $A_1,A_2,\cdots,A_n$ are non singular