Math 1600B Lecture 13, Section 2, 3 Feb 2014

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Announcements:

Read Sections 3.1 and 3.2 for next class. Work through recommended homework questions.

Quiz 4 is this week, and will focus on the material in Section 2.3 (linear (in)dependence), 2.4 (networks) and part of 3.1 (what we cover today).

Office hour: today, 1:30-2:30, MC103B.

Help Centers: Monday-Friday 2:30-6:30 in MC 106.

Lecture 12:

We covered network analysis and electrical networks in Section 2.4. Since the material won't be used today, I won't summarize it. I didn't quite finish Exercise 2.4.20, so I leave it as an exercise to solve the system I derived on the board: $$ \begin{aligned} \ph I_1 + 2 I_2 \phantom{{}+ 4 I_3} &= 5 \\ 2 I_2 + 4 I_3 &= 8 \\ \ph I_1 - \ph I_2 + \ph I_3 &= 0 \end{aligned} $$

We aren't covering Section 2.5.

New material: Section 3.1: Matrix Operations

(Lots of definitions, but no tricky concepts.)

Definition: A matrix is a rectangular array of numbers called the entries. The entries are usually real (from $\R$), but may also be complex (from $\C$).

Examples:

$$ \kern-8ex \small % The Rules/struts create some space below the matrices: \mystack{ A = \bmat{ccc} 1 & -3/2 & \pi \\ \sqrt{2} & 2.3 & 0 \emat_\strut %\Rule{0pt}{0pt}{15pt} }{\qquad 2 \times 3} %\qquad %\mystack{ %\bmat{rr} % 1 & 2 \\ 3 & 4 %\emat %\Rule{0pt}{0pt}{10pt} %}{\mystack{\strut 2 \times 2}{\textbf{square}}} \qquad \mystack{ \bmat{rrrr} 1 & 2 & 3 \emat_\strut %\Rule{0pt}{60pt}{0pt} }{\mystackthree{1 \times 3}{\textbf{row matrix}}{\text{or }\textbf{row vector}}} \qquad \mystack{ \bmat{r} 1 \\ 2 \\ 3 \emat_\strut }{\mystackthree{3 \times 1}{\textbf{column matrix}}{\text{or }\textbf{column vector}}} $$ The entry in the $i$th row and $j$th column of $A$ is usually written $a_{ij}$ or sometimes $A_{ij}$. For example, $$A_{11} = 1, \quad A_{23} = 0, \quad A_{32} \text{ doesn't make sense} . $$

Definition: An $m \times n$ matrix $A$ is square if $m = n$. The diagonal entries are $a_{11}, a_{22}, \ldots$. If $A$ is square and the nondiagonal entries are all zero, then $A$ is called a diagonal matrix. $$ % The Rules create some space below the matrices: \kern-8ex \small \mystack{ \bmat{ccc} 1 & -3/2 & \pi \\ \sqrt{2} & 2.3 & 0 \emat \Rule{0pt}{0pt}{18pt} }{\text{not square or diagonal}} \qquad \mystack{ \bmat{rr} 1 & 2 \\ 3 & 4 \emat \Rule{0pt}{0pt}{22pt} }{\text{square}} \qquad \mystack{ \bmat{rr} 1 & 0 \\ 0 & 4 \emat \Rule{0pt}{0pt}{20pt} }{\text{diagonal}} \qquad \mystack{ \bmat{rr} 1 & 0 \\ 0 & 0 \emat \Rule{0pt}{0pt}{20pt} }{\text{diagonal}} $$

Definition: A diagonal matrix with all diagonal entries equal is called a scalar matrix. A scalar matrix with diagonal entries all equal to $1$ is an identity matrix. $$ \kern-8ex \small % The Rules create some space below the matrices: \mystack{ \bmat{rr} 2 & 0 \\ 0 & 2 \emat \Rule{0pt}{0pt}{20pt} }{\text{scalar}} \qquad \mystack{ I_3 = \bmat{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \emat \Rule{0pt}{0pt}{18pt} }{\text{identity matrix}} \qquad \mystack{ O = \bmat{rrr} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \emat \Rule{0pt}{0pt}{18pt} }{\text{scalar}} $$ Note: Identity $\implies$ scalar $\implies$ diagonal $\implies$ square.

Now we're going to mimick a lot of what we did when we first introduced vectors.

Definition: Two matrices are equal if they have the same size and their corresponding entries are equal. $$ \kern-8ex \small % The Rules create some space below the matrices: \bmat{cc} 1 & -3/2 \\ \sqrt{2} & 0 \emat \quad \bmat{cc} \cos 0 & -1.5 \\ \sqrt{2} & \sin 0 \emat \quad \bmat{cc} 1 & 2 \\ 3 & 4 \emat \quad \bmat{rrrr} 1 & 2 & 3 & 4 \emat \quad \bmat{r} 1 \\ 2 \\ 3 \\ 4 \emat $$ The first two above are equal, but no other two are equal. We distinguish row matrices from column matrices!

Matrix addition and scalar multiplication

Our first two operations are just like for vectors:

Definition: If $A$ and $B$ are both $m \times n$ matrices, then their sum $A + B$ is the $m \times n$ matrix obtained by adding the corresponding entries of $A$ and $B$. Using the notation $A = [a_{ij}]$ and $B = [b_{ij}]$, we write $$ A + B = [a_{ij} + b_{ij}] \qquad\text{ or }\qquad (A+B)_{ij} = a_{ij} + b_{ij} . $$

Examples: $$ \bmat{rrr} 1 & 2 & 3 \\ 4 & 5 & 6 \emat + \bmat{rrr} 0 & -1 & 2 \\ \pi & 0 & -6 \emat = \bmat{ccc} 1 & 1 & 5 \\ 4+\pi & 5 & 0 \emat $$ $$ \bmat{rrr} 1 \\ 4 \emat + \bmat{rrr} 0 & -1 \emat \qtext{is not defined} $$

Definition: If $A$ is an $m \times n$ matrix and $c$ is a scalar, then the scalar multiple $cA$ is the $m \times n$ matrix obtained by multiplying each entry by $c$. We write $cA = [c \, a_{ij}]$ or $(cA)_{ij} = c \, a_{ij}$.

Example: $$ 3 \bmat{rrr} 0 & -1 & 2 \\ \pi & 0 & -6 \emat = \bmat{rrr} 0 & -3 & 6 \\ 3 \pi & 0 & -18 \emat $$

Definition: As expected, $-A$ means $(-1)A$ and $A-B$ means $A + (-B)$.

The $m \times n$ zero matrix has all entries $0$ and is denoted $O$ or $O_{m\times n}$. Of course, $A + O = A$.

So we have the real number $0$, the zero vector $\vec 0$ (or $\boldsymbol{0}$ in the text) and the zero matrix $O$.

 

Matrix multiplication

This is unlike anything we have seen for vectors.

Definition: If $A$ is $m \times \red{n}$ and $B$ is $\red{n} \times r$, then the product $C = AB$ is the $m \times r$ matrix whose $i,j$ entry is $$ c_{ij} = a_{i\red{1}} b_{\red{1}j} + a_{i\red{2}} b_{\red{2}j} + \cdots + a_{i\red{n}} b_{\red{n}j} = \sum_{\red{k}=1}^{n} a_{i\red{k}} b_{\red{k}j} . $$ This is the dot product of the $i$th row of $A$ with the $j$th column of $B$.

Note that for this to make sense, the number of columns of $A$ must equal the number of rows of $B$. $$ \mystack{A}{m \times n} \ \ \mystack{B}{n \times r} \mystack{=}{\strut} \mystack{AB}{m \times r} $$ This may seem very strange, but it turns out to be useful. We will never use componentwise multiplication, as it is not generally useful.

Examples on board: $2 \times 3$ times $3 \times 4$, $1 \times 3$ times $3 \times 1$, $3 \times 1$ times $1 \times 3$.

One motivation for this definition of matrix multiplication is that it comes up in linear systems.

Example 3.8: Consider the system $$ \begin{aligned} 4 x + 2 y &= 4 \\ 5 x + \ph y &= 8 \\ 6 x + 3 y &= 6 \end{aligned} $$ The left-hand sides are in fact a matrix product: $$ \bmat{rr} 4 & 2 \\ 5 & 1 \\ 6 & 3 \emat \coll x y $$ Every linear system with augmented matrix $[A \mid \vb\,]$ can be written as $A \vx = \vb$.

Note: In general, if $A$ is $m \times n$ and $B$ is a column vector in $\R^n$ ($n \times 1$), then $AB$ is a column vector in $\R^m$ ($m \times 1$). So one thing a matrix $A$ can do is transform column vectors into column vectors. This point of view will be important later.

Question: If $A$ is an $m \times n$ matrix and $\ve_1$ is the first standard unit vector in $\R^n$, what is $A \ve_1$?

Powers

In general, $A^2 = AA$ doesn't make sense. But if $A$ is $n \times n$ (square), then $A^2 = AA$ does make sense. $A^2$ is $n \times n$ as well, and so it also makes sense to define the power $$A^k = AA\cdots A \quad\text{with $k$ factors}.$$

We write $A^1 = A$ and $A^0 = I_n$.

We will see later that $(AB)C = A(BC)$, so the expression for $A^k$ is unambiguous. And it follows that $$ A^r A^s = A^{r+s} \qquad\text{and}\qquad (A^r)^s = A^{rs} $$ for all nonnegative integers $r$ and $s$.

Example 3.13 on board: Powers of $$ A = \bmat{rr} 1 & 1 \\ 1 & 1 \emat $$

True/false: Every diagonal matrix is a scalar matrix.

True/false: If $A$ is diagonal, then so is $A^2$.

True/false: If $A$ and $B$ are both square, then $AB$ is square.

Challenge question (for next class): Is there a nonzero matrix $A$ such that $A^2 = O$?

Next class: We'll cover the properties these operations have, from Section 3.2.