1. Mind Map

2. Reading Notes (2.4-2.5)

2.4 Rules for Matrix Operations

4 ways to find AB. (matrix multiplication)

• Take the dot product of each row of A with each column of B.
• Each column of AB is a combination of the columns of A. (Matrix A times every column of B)
• Every row of AB is a combination of the rows of B. (Every row of A times matrix B)
• Multiply columns 1 to n of A times rows 1 ton of B. Add those matrices.

The Laws for Matrix Operations

• Addition Law
  • commutative law
  • distributive law
  • associative law
• Multiplication Law
  • associative law for ABC
  • distributive law from the left
  • distributive law from the right
• Law of exponents
  • $A^{P} = AAA\cdots A\text{(p factors)}$
  • $(A^p)(A^q)=A^{p+q}$
  • $(A^p)^q=A^{pq}$

Block Matrices and Block Multiplication

• $\begin{bmatrix} I & 0 \cr -CA^{-1} & I \end{bmatrix} \begin{bmatrix} A & B \cr C & D \end{bmatrix} = \begin{bmatrix} A & B \cr 0 & D-CA^{-1}B \end{bmatrix}$
• Block elimination produces the Schur complement $D-CA^{-1}B$.

2.5 Inverse Matrices

• Two sided inverse $A^{-1}A = AA^{-1}=I$
• IMPORTANT: If A is invertible, then Ax = 0 can only have the zero solution $x = A^{-1}0 = 0$.
• OR: If Ax = 0 for a nonzero vector x, then A has no inverse.

The Inverses of a Product AB

If A and B are invertible then so is AB. Then inverse of a product AB is

$(AB)^{-1}=B^{-1}A^{-1}$

• Inverse of AB
• $(AB)B^{-1}A^{-1} = AIA^{-1} = AA^{-1}=I$
• $(ABC)^{-1}= C^{-1}B^{-1}A^{-1}$

Inverse of an elimination matrix

$E=\begin{bmatrix}1&0&0 \cr -5 & 1 & 0 \cr 0 & 0 & 1 \end{bmatrix} \quad and \quad E^{-1} =\begin{bmatrix}1&0&0 \cr 5 & 1 & 0 \cr 0 & 0 & 1 \end{bmatrix}$

For square matrices, an inverse on one side is automatically an inverse on the other side.

$F=\begin{bmatrix}1&0&0 \cr -0 & 1 & 0 \cr 0 & -4 & 1 \end{bmatrix} \quad and \quad F^{-1} =\begin{bmatrix}1&0&0 \cr 0 & 1 & 0 \cr 0 & 4 & 1 \end{bmatrix}$

• In elimination order F follows E. In reverse order $E^{-1}$ follows $F^{-1}$.
• $FE=\begin{bmatrix}1&0&0 \cr -5 & 1 & 0 \cr 20 & -4 & 1 \end{bmatrix} \quad \text{is inverted by } E^{-1}F^{-1} =\begin{bmatrix}1&0&0 \cr 5 & 1 & 0 \cr 0 & 4 & 1 \end{bmatrix}$
• $E^{-1}F^{-1}$ is quick. The multipliers 5, 4 fall into place below the diagonal of 1 ’s.

Calculating $A^{-1}$ by Gauss-Jordan Elimination

Each of the columns $x_1,x_2,x_3$ of $A^{-1}$ is multiplied by $A$ to produce a column of $I$:

• $AA^{-1} = A\begin{bmatrix} x_1&x_2&x_3\end{bmatrix} = \begin{bmatrix} e_1&e_2&e_3\end{bmatrix} = I$
• $Ax_1=e_1=(1,0,0);Ax_2=e_2;\cdots$

The Gauss-Jordan method computes $A^{-1}$ by solving all n equations together.

• $\begin{bmatrix} K & e_1 & e_2 & e_3 \end{bmatrix} =\begin{bmatrix} K & I \end{bmatrix} \Rightarrow \begin{bmatrix} I & x_1 & x_2 & x_3 \end{bmatrix} =\begin{bmatrix} I & K^{-1} \end{bmatrix}$

Singular verus Invertible

• $A^{-1} \text{ exists exactly when A has a full set of n pivots.(Row exchange allowed)}$
• A triangular matrix is invertible if and only if no diagonal entries are zero.

Recognizing an Invertible Matrix

Diagonally dominant matrices are invertible.（Unnecessary and sufficient condition）

• Diagonally dominant matrices are invertible.
• $|a_{ii}| > \sum_{j\neq i}|a_{ij}|$
• $\text{Example: } \begin{bmatrix} 3 & 1 & 1 \cr 1 & 3 & 1 \cr 1 & 1 & 3 \end{bmatrix}$