Section: 显示数据的矩阵 | 8.3 显示数据的矩阵

Section outline

A matrix is a rectangular array of numbers representing data in a variety of forms. Computers work very heavily with matrices because operations with matrices are efficient with memory. Matrices can represent statistical data with numbers, but also graphical data with pictures.
::矩阵是一个代表不同形式数据的矩形数字阵列,计算机对矩阵作用很大,因为矩阵操作对内存有效。矩阵可以代表带有数字的统计数据,也可以代表带有图片的图形数据。

How might you use a matrix to write the following image as something a computer could recognize and work with?
::您如何使用矩阵写下以下图像作为计算机能够识别和工作的东西 ?

Introduction to Matrices
::数学入门

A matrix is a means of storing information effectively and efficiently. The rows and columns each mean something very specific and the location of a number is just as important as its value. The following are all examples of matrices:
::矩阵是有效和高效地储存信息的一种手段,各行和列意味着非常具体的东西,数字的位置与其价值同样重要。

$\begin{aligned} [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix}], [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}], [\begin{matrix} 2 & 3 & 4 & 5 \\ 0 & 4 & 5 & 0 \\ 0 & 0 & 2 & - 9 \\ 0 & 0 & 0 & 10 \end{matrix}] \end{aligned}$

The entries in a matrix can be written out using brackets like [ ], but they can also be described individually using a set of 2 subscript indices $\begin{aligned} i \end{aligned}$ and $\begin{aligned} j \end{aligned}$ that stand for the row number and the column number. Alternatively, the matrix can be named with just a capital letter like $\begin{aligned} A \end{aligned}$ .
::矩阵中的条目可以用括号如[ ]来写出,但也可以用一组2个下标指数i和j来单独描述,它们代表行号和列号。或者,矩阵可以仅用象A这样的大写字母命名。

$\begin{aligned} A = [a_{i j}] = [\begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}] \end{aligned}$
::A=[aij]=[a11a12a21a22]

Square matrices have the same number of rows as columns. The order of a matrix describes the number of rows and the number of columns in the matrix. The following matrix is said to have order $\begin{aligned} 2 \times 3 \end{aligned}$ because it has two rows and three columns. A $\begin{aligned} 1 \times 1 \end{aligned}$ matrix is just a regular number.
::方块矩阵的行数与列数相同。矩阵的顺序说明矩阵中的行数和列数。以下的矩阵据说有顺序2×3,因为它有两个行和三列。A 1×1矩阵只是一个普通数字。

$\begin{aligned} [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix}] \end{aligned}$

A symmetric matrix is a special type of square matrix that has reflection across the main diagonal. The identity matrix is an example of a symmetric matrix.
::对称矩阵是一种特殊的平方矩阵类型,在主对角反射,身份矩阵是对称矩阵的一个例子。

The identity matrix of order $\begin{aligned} n \times n \end{aligned}$ has zeros everywhere except along the main diagonal where it has ones. Just like the number 1 has an important property with numbers, the identity matrix of any order has special properties as well.
::Nxn 顺序的身份矩阵除主对角有的外,无处是零。正如数字 1 具有重要的带有数字的属性一样,任何顺序的身份矩阵也具有特殊属性。

$\begin{aligned} [1], [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}], [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] \end{aligned}$

When you turn the rows of a matrix into the columns of a new matrix, the two matrices are transpositions of one another. The superscript $\begin{aligned} T \end{aligned}$ stands for transpose. Sometimes using the transpose of a matrix is more useful than using the matrix itself.
::当您将矩阵的行转换为新矩阵的列时,两个矩阵是相互转换的。上标 T 代表转换。有时,使用矩阵的转换比使用矩阵本身更有用。

\begin{aligned} A & = [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix}] \\ A^{T} & = [\begin{matrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{matrix}] \end{aligned}

::A=[123456]AT=[142536]

A triangular matrix is not a matrix in the shape of a triangle. Rather, a lower triangular matrix is a square matrix where every entry below the diagonal is zero. An upper triangular matrix is a square matrix where every entry above the diagonal is zero. The following is a lower triangular matrix. When you work with solving matrices, look for triangular matrices because they are much easier to solve.
::三角矩阵不是三角形的矩阵。相反,下三角矩阵是一个方格矩阵,对角线下每个条目为零。上三角矩阵是一个方格矩阵,对角线上每个条目为零。下三角矩阵是一个较低三角矩阵。当您使用解析矩阵时,请寻找三角矩阵,因为三角矩阵更容易解答。

$\begin{aligned} [\begin{matrix} 2 & 3 & 4 & 5 \\ 0 & 4 & 5 & 0 \\ 0 & 0 & 2 & - 9 \\ 0 & 0 & 0 & 10 \end{matrix}] \end{aligned}$

A diagonal matrix is both upper and lower triangular which means all the entries except those along the diagonal are zero. The identity matrix is a special case of a diagonal matrix.
::对角矩阵为上三角形和下三角形,这意味着除对角矩阵外的所有条目均为零。身份矩阵是对角矩阵的一个特例。

Examples
::实例

Example 1
::例1

Earlier, you were asked how you might use a matrix to write the following image as something a computer could recognize and work with.
::早些时候,有人问您如何使用矩阵写下以下图像,作为计算机能够识别和工作的东西。

By writing every hollow square as a 0 and a blank square as a 1 a computer could read the picture:
::将每个空方块写成0,将空方块写成1,计算机就能读出图片:

When you use computers to manipulate images, the computer just manipulates the numbers. In this case, if you swap zeros and ones, you get the negative image.
::当您使用计算机来操作图像时, 计算机只会操纵数字。在这种情况下, 如果您交换零和一, 您就会得到负图像。

Real photos and computer images have matrices that are much larger and include more numbers than just zero and one to account for more colors.
::真实照片和计算机图像的矩阵要大得多,包含的数字多于零和一,以说明更多颜色。

Example 2
::例2

Kate runs three bakeries and each bakery sells bagels and muffins. The rows represent the bakeries and the columns represent bagels (left) and muffins (right) sold. Answer the following questions about Kate’s sales.
::凯特经营三家面包店,每家面包店出售百吉饼和松饼。面包行代表面包店,各栏代表百吉饼(左)和松饼(右)。回答以下关于凯特销售的问题。

\begin{aligned} K = [\begin{matrix} 144 & 192 \\ 115 & 127 \\ 27 & 34 \end{matrix}] \end{aligned}

::K=[1441921151272734]

What does 127 represent?
1. 127 represents the number of muffins that Kate sold in her second location. You know this because it is in the muffin column and the second row.
  ::127 表示凯特在第二处卖出的松饼数量。你知道这一点,因为它在松饼栏和第二排。
::127代表什么? 127代表凯特在第二个地点出售的松饼数量。你知道这一点,因为它在松饼栏和第二排。

How many muffins did Kate sell in total?
1. The total muffins sold is equal to the sum of the right hand column. $\begin{aligned} 192 + 127 + 34 = 353 \end{aligned}$
  ::出售的松饼总数等于右手栏192+127+34=353的总和。
::Kate总共卖了多少松饼? 销售的松饼总数等于右手栏192+127+34=353的总和。

How many bagels did Kate sell in her first location?
1. Kate sold 144 bagels at her first location.
  ::Kate在第一地点卖了144个百吉饼
::凯特在第一个地方卖了多少百吉饼?

Which location is doing poorly?
1. The third location is doing much worse than the other two locations.
  ::第三个地点比其他两个地点糟糕得多。
::哪个位置不好?第三个位置比其他两个位置差得多。

Example 3
::例3

Identify the order of the following matrices
::确定下列矩阵的顺序

\begin{aligned} A = [\begin{matrix} 1 & 3 & 4 & 7 \end{matrix}], B = [\begin{matrix} 21 & 45 & 1 \\ 34 & 1 & 5 \end{matrix}], C = [\begin{matrix} 25 & 235 \\ 562 & 562 \\ 4 & 413 \\ 454 & 33 \\ 1 & 141 \end{matrix}] \end{aligned}

::A=[1347],B=[214513415],C=[25235556256224413434331141]

$\begin{aligned} A \end{aligned}$ is $\begin{aligned} 1 \times 4 \end{aligned}$ , $\begin{aligned} B \end{aligned}$ is $\begin{aligned} 2 \times 3 \end{aligned}$ , $\begin{aligned} C \end{aligned}$ is $\begin{aligned} 5 \times 2 \end{aligned}$ . Note that $\begin{aligned} 4 \times 1, 3 \times 2, 2 \times 5 \end{aligned}$ are not the same orders and would be incorrect.
::A 是 1x4, B 是 2x3, C 是 5x2. 注意 4x1, 1x2, 2x5 并非同一订单, 且不正确。

Example 4
::例4

Write out the $\begin{aligned} 4 \times 5 \end{aligned}$ matrix whose entries are $\begin{aligned} a_{i j} = \frac{i + j}{j} \end{aligned}$ .
::填写 Aij=i+jj 条目的 4x5 矩阵。

\begin{aligned} [\begin{matrix} 2 & \frac{3}{2} & \frac{4}{3} & \frac{5}{4} & \frac{6}{5} \\ 3 & 2 & \frac{5}{3} & \frac{3}{2} & \frac{7}{5} \\ 4 & \frac{5}{2} & 2 & \frac{7}{4} & \frac{8}{5} \\ 5 & 3 & \frac{7}{3} & 2 & \frac{9}{5} \end{matrix}] \end{aligned}

Example 5
::例5

Create a $\begin{aligned} 3 \times 3 \end{aligned}$ matrix for each of the following:
::为以下每一种创建 3x3 矩阵 :

a. Diagonal Matrix
::a. 对角矩阵

Possible answer:
::可能的答复:

$\begin{aligned} [\begin{matrix} 4 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{matrix}] \end{aligned}$

b. Lower Triangular
::b. 下三角三角形

Possible answer:
::可能的答复:

$\begin{aligned} [\begin{matrix} 4 & 1 & 1 \\ 0 & 3 & 14 \\ 0 & 0 & 5 \end{matrix}] \end{aligned}$

c. Symmetric
::c. 对称

Possible answer:
::可能的答复:

$\begin{aligned} [\begin{matrix} 4 & 1 & 1 \\ 1 & 3 & 14 \\ 1 & 14 & 5 \end{matrix}] \end{aligned}$

d. Identity:
::d. 身份:

Answer:
::答复:

$\begin{aligned} [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] \end{aligned}$

Note that while the identity matrix does technically work for all the parts of this problem, it does not highlight the differences between each definition.
::请注意,虽然身份信息总库在技术上对该问题的所有部分都起作用,但它没有突出每个定义之间的差别。

Summary

A matrix is a rectangular array of numbers representing data, used for efficient storage and manipulation of information in rows and columns.
::矩阵是一个代表数据的矩形数字阵列,用于在行和列中有效储存和操作信息。

Square matrices have the same number of rows and columns,
::平方矩阵的行数和列数相同,

The order of a matrix describes the number of rows and columns in the matrix.
::矩阵的顺序说明矩阵中的行数和列数。

A symmetric matrix is a square matrix with reflection symmetry across the main diagonal.
::对称矩阵是一个平方矩阵,在主对角上具有反射对称。

The identity matrix is an $\begin{aligned} n \times n \end{aligned}$ matrix that has zeros everywhere except along the main diagonal which has ones.
::身份矩阵是一个nxn矩阵,除主要对角有零位数外,无处不在。

Transpositions occur when the rows of a matrix are turned into the columns of a new matrix, denoted by the superscript T.
::当将矩阵的行转换成由上标 T 表示的新矩阵的列时,就会发生转换。

Triangular matrices are square matrices with either all entries below the diagonal (lower triangular) or all entries above the diagonal (upper triangular) being zero.
::三角矩阵是方格矩阵,所有条目要么在对角(下三角)下,要么在对角(上三角)上方的所有条目为零。

Diagonal matrices have all entries except those along the diagonal as zero.
::对角矩阵除沿对角矩阵的条目为零外,所有条目都有。

Review
::回顾

State the order of each of the following matrices:
::列出下列表格的顺序:

1. $\begin{aligned} A = [\begin{matrix} 4 & 2 & 4 & 7 \\ 5 & 2 & 1 & 0 \end{matrix}] \end{aligned}$
::1. A=[42475210]

2. $\begin{aligned} B = [\begin{matrix} 0 & 1 \\ 34 & 1 \end{matrix}] \end{aligned}$
::2. B=[01341]

3. $\begin{aligned} C = [\begin{matrix} 2 & 62 \\ 14 & 3 \\ 4 & 3 \\ 1 & 11 \end{matrix}] \end{aligned}$
::3. C=[2624143111]

4. $\begin{aligned} D = [\begin{matrix} 12 & 0 & 2 \\ 0 & 3 & 3 \\ 4 & 0 & 1 \\ 1 & 4 & 0 \end{matrix}] \end{aligned}$
::4. D=[12033401140]

5. $\begin{aligned} E = [\begin{matrix} 1 & 11 \end{matrix}] \end{aligned}$
::5. E=[111]

6. Give an example of a $\begin{aligned} 1 \times 1 \end{aligned}$ matrix.
::6. 举例说明1x1矩阵。

7. Give an example of a $\begin{aligned} 3 \times 2 \end{aligned}$ matrix.
::7. 举一个3x2矩阵的例子。

8. If a symmetric matrix is also lower triangular, what type of matrix is it?
::8. 如果对称矩阵也是较低三角形的,那么它是什么类型的矩阵?

9. Write out the $\begin{aligned} 2 \times 3 \end{aligned}$ matrix whose entries are $\begin{aligned} a_{i j} = i - j \end{aligned}$ .
::9. 填写条目为aij=i-j的2×3矩阵。

Morgan worked for three weeks during the summer earning money on Mondays, Tuesdays, Wednesdays, Thursdays, and Fridays. The following matrix represents his earnings.
::Morgan在夏季工作了三周,在星期一、星期二、星期三、星期四和星期五挣钱。下面的表格代表他的收入。

$\begin{aligned} [\begin{matrix} 24 & 22 & 32 \\ 25 & 28 & 30 \\ 30 & 28 & 32 \\ 10 & 15 & 19 \\ 35 & 32 & 30 \end{matrix}] \end{aligned}$

10. What do the rows and columns represent?
::10. 列和列代表什么?

11. How much money did Morgan make in the first week?
::11. Morgan在第一周赚了多少钱?

12. How much money did Morgan make on Tuesdays?
::12. 摩根星期二赚了多少钱?

13. What day of the week was most profitable?
::13. 星期的哪一天最有利可图?

14. What day of the week was least profitable?
::14. 星期的哪一天利润最低?

15. Is the following a matrix? Explain.
::15. 以下是矩阵表吗?

$\begin{aligned} [\begin{matrix} d o g s & 0 \\ c a t s & 3 \\ s h e e p & 0 \\ d u c k s & 4 \end{matrix}] \end{aligned}$
::[狗猫3she0ducks4] [狗猫3she0ducks4]

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

Section outline

Introduction to Matrices ::数学入门

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Introduction to Matrices
::数学入门

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)