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 \(A^{-1}b\) 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
