MIT18.06學習筆記 - Lecture 2: Elimination with Matrices

這個系列文章是我重溫Gilbert老爺子的線性代數在線課程的學習筆記。
Course Name：MIT 18.06 Linear Algebra
Text Book: Introduction to Linear Algebra
章節內容: 2.2-2.3

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課程提綱
1. Elimination and Back-Substitution
2. Elimination Matrix E and Permutation Matrix P
3. The Augmented Matrix

課程重點
Gaussian elimination (also known as row reduction) is an algorithm for solving systems of linear equations. It is usually understood as a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix.

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Elimination and Back-Substitution

The systematic way to solve linear equations: elimination, 所有軟件如matlab的求解矩陣的方法。
Elimination produces an upper triangular system，and use back substitution to solve it:

The word “entry” for a matrix corresponds to “component” for a vector. General rule: aij=A(i,j)$a_{ij}=A(i,j)$ is in row i$i$ , column j$j$ .
To perform Gaussian Elimination (row reduction) on a matrix, one uses a sequence of elementary row operations to modify the matrix until the lower left-hand corner of the matrix is filled with zeros, as much as possible. There are three types of elementary row operations:

• Swapping two rows,
• Multiplying a row by a nonzero number,
• Adding a multiple of one row to another row.

The pivots are on the diagonal of the triangle after elimination.

Failure, breakdown of elimination: For n$n$ equations we do not get n$n$ pivots.
Elimination leads to an equation 0≠0$0\ne 0$ (no solution) and 0=0$0=0$ (many solutions).
Success comes with n$n$ pivots. But we may have to exchange the n$n$ equations.

Elimination Matrix E and Permutation Matrix P

The Matrix Eij$E_{ij}$ for One Elimination Step
Multiplication by elimination matrix E21$E_{21}$ subtracts 2b1$2b_1$ from b2$b_2$ and Rows 1 and 3 stay same:

The identity matrix has 1’s on the diagonal and otherwise 0’s. Then Ib=b$Ib=b$ for all b$b$ . The elimination matrix Eij$E_{ij}$ that subtracts a multiple l$l$ of row j$j$ from row i$i$ has the extra nonzero entry −l$-l$ in the i$i$ , j$j$ position (still diagonal 1’s).

The Matrix Pij$P_{ij}$ for a Row Exchange
Multiplying by permutation matrix P23$P_{23}$ exchanges rows 2 and 3 of any matrix:

Two Elimination Steps

The Augmented Matrix

Elimination does the same row operations to A$A$ and b$b$ , we can include b$b$ as an extra column and follow it through elimination: