Determinats(行列式) 2018-11-23

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 on each element of b. If one parameter is changed in an experiment, or one observation is corrected, the "influence coefficient" in 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 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:
  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(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:
  The cofactor is the determinant of with the correct sign:
  delete row i and column j
  These formulas express detA as a combination of determinants of order n-1

4. Applications of Determinants

4.1 Computation of

• Cofactor matrix, C is transposed
  means

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

The jth component of is the ratio
where (has b in column j)

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 . For every k, the submatrix is going through a Gaussian elimination of its own.
• Formula for pivots: (By convention, )
  Mutiplying together all the individual pivots, we recover:
  
  The pivot entries are all nonzero whenever the number are all nonzero
• Elimination can be completed without row exchanges (so P=I and A = LU), if and only if the leading submatrices are non singular