Linear Algebra : Chapter 2

Chapter 2 Matrix Algebra

Section 2.1 Matrix Operations

• diagonal entries : $a_{1,1},a_{2,2},\dots$
  • diagonal matrix : all nondiagonal entries are all zero
• Equal : two matrix are equal iff they have same size
• scalar multiple : $rA$

Matrix Multiplication

If $A$ is an $m\times n$ matrix, and if $B$ is an $n\times p$ matrix with columns $b_1,\dots,b_p$, then the product $AB$ is the $m\times p$ matrix whose columns are $Ab_1,\dots,Ab_p$. That is,

$$AB=A[b_1\;b_2\;\cdots b_p]=[Ab_1\;Ab_2\cdots Ab_p]$$

• $A(Bx)=(AB)x$

Notice that since the first column of $AB$ is $Ab_1$, this column is a linear combination of the columns of $A$ using the entries in $b_1$ as weights. A similar statement is true for each column of $AB$

The columns of $AB$ are linear combinations of the columns of $A$

Row-Column Rule For Computing AB :

$$(AB)_{ij}=a_{i1}b_{1j}+\dots+a_{in}b_{nj}$$

Also, we have:

$$\text{row}_i(AB)=\text{row}_i(A)\cdot B$$

I think this is also true for column(as the definition)

Properties:

If $AB=BA$, we say $A$ and $B$ are commute with one another.

Warning :

1. In general, $AB\neq BA$
2. If $AB= AC$, then in general $B\neq C$
3. If $AB=0$, you can not conclude $A=0$ or $B=0$

I think if all the columns of $A$ are linearly independent, then we know $B=C$

Power of Matrix

Transpose of a Matrix

The transpose of a $m\times n$ matrix $A$, is a $n\times n$ matrix, denoted by $A^T$, whose columns are formed from the corresponding rows of $A$.

Properties:

• $(A^T)^T=A$
• $(A+B)^T=A^T+B^T$
• $(rA)^T=rA^T$
• $(AB)^T=B^TA^T$ (Important)