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Determinats(行列式) 2018-11-23

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 givesdet 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

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