MIT18.06學習筆記 - Lecture 3: Multiplication and Inverse 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: 2.4-2.5

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課程提綱
1. Matrix Multiplication (5 ways)
2. The laws for Matrix Operations
3. Inverse Matrices
4. Gauss-Jordan Elimination

課程重點


Row echelon form resulting from Gaussian elimination: an upper triangular matrix U$U$ .
Reduced row echelon form computed by Gauss–Jordan elimination: all of the leading coefficients (the leftmost nonzero entry in each row) are 1, and every column containing a leading coefficient has zeros elsewhere, it is unique and independent of the sequence of row operations used. For example, the third and fourth matrices are the ones in row echelon form, and the final matrix is the unique reduced row echelon form:

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Matrix Multiplication (5 ways)

1. Entry view, the entry in row i$i$ and column j$j$ of AB$AB$ is (row i$i$ of A$A$ ) ⋅$\cdot$ (column j$j$ of B$B$ ):

2. Column view, columns of C$C$ are combinations of columns of A:

3. Row view, rows of C$C$ are combinations of rows of B:

4. Outer product view, AB$AB$ adds up columns times rows, important special case of block view:

Compare it with the usual rows times columns. Row 1 of A$A$ times column 1 of B$B$ gave the (1,1) entry in AB$AB$ . Now column 1 of A$A$ times row 1 of B$B$ gives a full matrix - not just a single number:

5. Block view, matrix can be cut into blocks, if the cuts between columns of A$A$ match the cuts between rows of B$B$ , block multiplication of AB$AB$ is allowed:

Block matrix can do elimination on a whole (block) column of A$A$ :

This is the ordinary elimination, a column at a time - written in blocks. That final block S$S$ is called the Schurcomplement$Schurcomplement$ .

The laws for Matrix Operations

Inverse matrix is A−1$A^{-1}$ and A0=I$A^0=I$ .

Inverse Matrices

The matrix A$A$ is invertible if there exists a matrix A−1$A^{-1}$ such that A−1A=I$A^{-1}A=I$ and AA−1=I$A{A}^{-1}=I$ . Six notes about A−1$A^{-1}$ :
Note 1: The inverse exists if and only if elimination products n$n$ pivots (row exchanges are allowed).
Note 2: Suppose BA=I$BA=I$ and also AC=I$AC=I$ , then B=C$B=C$ , according to B(AC)=(BA)C$B(AC)=(BA)C$
Note 3: If A$A$ is invertible, the one and only solution to Ax=b$Ax=b$ is x=A−1B$x=A^{-1}B$
Note 4: Suppose there is a nonzero vector x$x$ such that Ax=0$Ax=0$ , then A$A$ cannot have an inverse.
Note 5: A 2 by 2 matrix is invertible if and only if ad−bc$ad-bc$ is not zero:

This number ad−bc$ad-bc$ is the determinant of A$A$ . A matrix is invertible if its determinant is not zero, but the test for n$n$ pivot is usually decided before the determinant appears.
Note 6: A diagonal matrix has an inverse provided no diagonal entries are zeros:

The Inverse of a product AB$AB$ comes in reverse order: (AB)−1=B−1A−1$(AB{)}^{-1}=B^{-1}{A}^{-1}$
For square matrices, an inverse on one side is automatically an inverse on the other side. If AB=I$AB=I$ then automatically BA=I$BA=I$ , where B=A−1$B=A^{-1}$ .

Gauss-Jordan Elimination

Th Gauss-Jordan method computes A−1$A^{-1}$ using block matrix [A I]$[AI]$ :

Here is an example, first part is doing Gaussian elimination to get U$U$ (upper triangular) in the first three columns, which is in Row Echelon Form:

Second part is to produce zeros above the pivots: Gauss would finish by back substitution but Jordan continue with elimination to the Reduced Echelon Form (after the last step):

The last step is to divide each row by its pivot, the first half of the result matrix is I$I$ , because K$K$ is invertible and the second half is K−1$K^{-1}$ :