Week 7 - More Gaussian Elimination and Matrix Inversion
When Gaussian Elimination Breaks Down
- Let and assume that completes with a matrix that has no zero elements on its diagonal.
- has a unique solution.
Let be a permutation vector. Then is said to be a permutation matrix.
If is a permutation matrix, then so is .
LU factorization algorithm that incorporates row (partial) pivoting
- Solving then changes to
- Compute such that .
- Update .
- Solve (forward substitution)
- Solve (backward substitution)
The Inverse Matrix
- Definition: an n-by-n square matrix A is called invertible (also nonsingular) if there exists an n-by-n square matrix B such that
Inverse Functions in 1D
- maps a rea to a real and it is a bijection(both one-to-one and onto)
- bijection means: every element in R, there is a unique output in R.
- has a unique solution for all .
- The function that maps y to x so that is called the inverse of .
- It is denoted by .
- If and , then , where is the th column of B and is the th unit basis vector.
Properties of the inverse
Assume A, B, and C are square matrices that are nonsingular. Then
The following statements are equivalent statements about :
- A is nonsingular(不可逆).
- A is invertible.
- undoes what matrix A did.
- Identity Matrix
- A represents a linear transformation that is a bijection.
- Ax = b has a unique solution for all .
- implies that .
- has a solution for all .
- The determinant of A is nonzero: . Will talk this in next section.
Theorem: Let be a permutation matrix. Then .
- The determinant of a matrix A is denoted
- So if , then doesn’t exist and is non-invertible.
- A formula for the determinant of a 3×3 matrix:
- Similar to the matrix.
Inverses of special matrices
- permutation [,pə:mju:'teiʃən] n. [数] 排列；[数] 置换
- bijection [bai’dʒekʃən] n. [数] 双射
- determinant [di’tə:minənt] adj. 决定性的 n. 决定因素；[数] 行列式