线性代数导论 (2-Solving Linear Equations)
Solving Linear Equations
Vectors and Linear Equations
The row picture shows tow lines meeting at a single point(solution).
The column picture combines the column vectors on the left side to product the vector $b$ on the right side.
The Matrix Form of the Equations $Ax=b$
Multiplication by rows : $Ax$ comes from dot products, each row times the column $x$ :
$$
Ax =
\begin{bmatrix}
(row1) \cdot x \
(row2) \cdot x \
(row3) \cdot x \
\end{bmatrix}
$$
Multiplication by columns $Ax$ is a combination of column vectors :
$$
Ax = x\ (column\ 1) + y\ (column\ 2) + z\ (column \ 3)
$$
消元的几何意义是坐标系的变换
The idea of Elimination
Pivot first nonzero in the row that does the elimination.
$$Ax=b \quad \Rightarrow \quad Ux=c $$
Elimination Using Matrices
$Ax$ is a combination of the column of $A$.
Components of $Ax$ are dot products with rows of $A$.
Matrix Multiplication
Associative law is true $A(BC)=(AB)C$
Commutative law is false $Often\ AB≠BA$
Permutation Matrix
$P_{ij}$ is the identify matrix with row $i$ and $j$ reversed.
Augmented Matrix
Augmented Matrix include $b$ as an extra column and follow it through elimination.
Rules for Matrix Operations
The entry in row $i$ and column $j$ is called $a_{ij}$ or $A(i, j)$
Rows and Columns of $AB$
Each column of $AB$ is a combination of the columns of $A$.
The Laws for Matrix Operations
$AB≠BA$ (the commutative “law” is usually broke)
$C(A + B) = CA + CB$ (distributive law from the left)
$(A + B)C = AC + BC$ (distributive law from the right)
$A(BC)=(AB)C$ (associative law for ABC)
Block Matrices and Block Multiplication
Suppose $A, B, C, D$ are respectively $p×p$, $p×q$, $q×p$ and $q×q$ matrices, and $D$ is invertible.
$$
\begin{bmatrix}
A & B \
C & D \
\end{bmatrix}
=
\begin{bmatrix}
I_p & BD^{-1} \
0 & I_q \
\end{bmatrix}
\begin{bmatrix}
A-BD^{-1}C & 0 \
0 & D \
\end{bmatrix}
\begin{bmatrix}
I_p & 0 \
D^{-1}C & I_q \
\end{bmatrix}
$$
the Schur complement of the block $D$ of the matrix $M$ is the $p×p$ matrix $A-BD^{-1}C$
Inverse Matrices
The matrix $A$ is invertible if there exists a matrix $A^{-1}$ such that $$A^{-1}A=I$$ and $$AA^{-1}=I$$
elimination produces n pivots $⇔$ the inverse exists $⇔$ non-singular matrix
The Inverse of a Product $AB$
$$(AB)^{-1}=B^{-1}A^{-1}$$
Gauss-Jordan Elimination
Multiply $\begin{bmatrix} A & I \end{bmatrix}$ by $A^{-1}$ to get $\begin{bmatrix} I & A^{-1} \end{bmatrix}$
A square matrix is called lower triangular matrix if all the entries above the main diagonal are zero. Similarly, a square matrix is called upper triangular matrix if all the entries below the main diagonal are zero.A triangular matrix is one that is either lower triangular or upper triangular. A matrix that is both upper and lower triangular is called a diagonal matrix.
A triangular matrix is invertible if and only if no diagonal entries are zero
Elimination = Factorization: $A = LU$
The factors $L$ and $U$ are triangular matrices. The factorization that comes from elimination is $A = LU$ or $A=LDU$. This is elimination without row exchanges.
Transposes and Permutations
The entry in row i, column j of AT comes from row j, column i of the original A : $(A^T){ij}=A{ji}$
- Sum
$$A+B=A^T+B^T$$
- Product
$$(AB)^T=B^TA^T$$
- Inverse
$$(A^{-1})^T=(A^{T})^{-1}$$
The Meaning of Inner Products
The dot product or inner product is $x^Ty$ (1 x n) (n x 1)
The rank one product or outer product is $xy^T$ (n x 1) (1 x n)
$x^Ty$ is a number, $xy^T$ is a matrix.
Symmetric Matrices
A symmetric matrix has $A^T=A$
The inverse of a symmetric matrix is also symmetric.
Symmetric Products $R^TR$ and $RR^T$ and $LDL^T$
The transpose of $R^TR$ is $R^T(R^T)^T$ which is $R^TR$
The symmetric factorization of a symmetric matrix is $A = LDL^T$
Permutation Matrices
A permutation matrix $P$ has the rows of the identify $I$ in any order.
$$P^T = P^{-1}$$
The $PA = LU$ Factorization with Row Exchanges
If $A$ is invertible then a permutation $P$ will reorder its rows for $PA = LU$
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