How to organize, add and multiply matrices - Bill Shillito

如何整理、相加、和相乘矩陣 - Bill Shillito

Translator: Andrea McDonough Reviewer: Bedirhan Cinar

譯者: Anny Chung 審譯者: Jephian Lin

By now, I'm sure you know

你大概知道

that in just about anything you do in life,

生活中幾乎任何工作

you need numbers.

都需要數字

In particular, though,

尤其是

some fields don't just need a few numbers,

有些領域不只需要一些數字

they need lots of them.

他們需要很多數字

How do you keep track of all those numbers?

要如何整理這麼多數字呢?

Well, mathematicians dating back

數學家早在古代中國

as early as ancient China

數學家早在古代中國

came up with a way to represent

就已發明用來同時表現

arrays of many numbers at once.

許多數字的辦法

Nowadays we call such an array a "matrix,"

現在我們叫這個組合方法「矩陣」

and many of them hanging out together, "matrices".

現在我們叫這個組合方法「矩陣」

Matrices are everywhere.

矩陣無所不在 （譯註：矩陣英文即電影駭客任務中的母體。）

They are all around us,

環繞我們四周

even now in this very room.

即使是現在，在這房間裡

Sorry, let's get back on track.

抱歉，我們回歸正題

Matrices really are everywhere, though.

矩陣真的無所不在

They are used in business,

它們被使用於商業

economics,

經濟

cryptography,

physics,

密碼學

物理

electronics,

電子

and computer graphics.

和電腦影像技術

One reason matrices are so cool

矩陣很酷的原因之一

is that we can pack so much information into them

是它們可以集中很多資訊

and then turn a huge series of different problems

並將一系列不同的問題

into one single problem.

整理成一個問題

So, to use matrices, we need to learn how they work.

使用矩陣前，我們需要了解它的運作方式

It turns out, you can treat matrices

其實，矩陣和普通數字差不多

just like regular numbers.

其實，矩陣和普通數字差不多

You can add them,

你可以將它們相加

subtract them,

相減

even multiply them.

或相乘

You can't divide them,

但你不能將它們相除

but that's a rabbit hole of its own.

那個課題有點複雜

Adding matrices is pretty simple.

矩陣相加很簡單

All you have to do is add the corresponding entries

你只要將其中的元素相加

in the order they come.

依照原來的順序

So the first entries get added together,

所以第一個元素相加

the second entries,

第二個元素

the third,

all the way down.

第三個

以此類推

Of course, your matrices have to be the same size,

當然，矩陣必須是相同的大小

but that's pretty intuitive anyway.

這應該不難理解

You can also multiply the whole matrix

你也可以將整個矩陣

by a number, called a scalar.

以一個數字，或純量，相乘

Just multiply every entry by that number.

只要將每個元素乘以那個數字即可

But wait, there's more!

但等等，還有更多！

You can actually multiply one matrix by another matrix.

你還可以將兩個矩陣相乘

It's not like adding them, though,

但是和相加不太一樣

where you do it entry by entry.

不是一個一個元素來

It's more unique

它比較特別

and pretty cool once you get the hang of it.

學會了之後其實挺有趣的

Here's how it works.

這就是它的原理

Let's say you have two matrices.

比如說你有兩個矩陣

Let's make them both two by two,

都是 2×2 的矩陣

meaning two rows by two columns.

也就是二列二行

Write the first matrix to the left

先將第一個矩陣寫在左邊

and the second matrix goes next to it

第二個矩陣在旁邊

and translated up a bit,

稍微偏上面

kind of like we are making a table.

像是在做表格一樣

The product we get when we multiply the matrices together

矩陣相乘的乘積

will go right between them.

將在這兩個矩陣中間

We'll also draw some gridlines to help us along.

我們來畫一些格線來做輔助

Now, look at the first row of the first matrix

先看看第一個矩陣的第一列

and the first column of the second matrix.

和第二個矩陣的第一行

See how there's two numbers in each?

看到它們各有兩個數字嗎？

Multiply the first number in the row

將第一行的第一個數字

by the first number in the column:

與第一列的第一個數字相乘

1 times 2 is 2.

1 乘 2 等於 2

Now do the next ones:

接下來將下一對相乘

3 times 3 is 9.

3 乘 3 等於 9

Now add them up:

接下來將它們相加

2 plus 9 is 11.

2 加 9 等於 11

Let's put that number in the top-left position

我們來把這個答案放在左上位置

so that it matches up with the rows and columns

讓它和我們剛剛利用的行列對齊

we used to get it.

讓它和我們剛剛利用的行列對齊

See how that works?

看清楚了嗎?

You can do the same thing to get the other entries.

你可以用相同方法得到其他的元素

-4 plus 0 is -4.

-4 加 0 是 -4

4 plus -3 is 1.

4 加 -3 是 1

-8 plus 0 is -8.

-8 加 0 是 -8

So, here's your answer.

所以這就是答案

Not all that bad, is it?

還不賴吧？

There's one catch, though.

但是有一個限制

Just like with addition,

就像是矩陣加法

your matrices have to be the right size.

你的矩陣必須是正確的大小

Look at these two matrices.

看看這兩個矩陣

2 times 8 is 16.

2 乘 8 是 16

3 times 4 is 12.

3 乘 4 是 12

3 times

3 乘

wait a minute,

等等

there are no more rows in the second matrix.

第二個矩陣中沒有更多列了

We ran out of room.

空間不夠

So, these matrices can't be multiplied.

所以這兩個矩陣無法相乘

The number of columns in the first matrix

第一個矩陣中的行數

has to be the same as the number of rows in the second matrix.

必須等於第二個矩陣中的列數

As long as you're careful

但是只要你小心這點

to match up your dimensions right, though,

確保行列數正確

it's pretty easy.

剩下的其實不難

Understanding matrix multiplication

學習矩陣乘法

is just the beginning, by the way.

只是個開始而已

There's so much you can do with them.

矩陣的用處還有更多

For example, let's say you want

比如說

to encrypt a secret message.

如果你想要傳遞秘密訊息

Let's say it's "Math rules".

像是「數學超棒（Math rules）」

Though, why anybody would want to keep this a secret

雖然我無法想像 為何有人會想把這當做秘密

is beyond me.

雖然我無法想像 為何有人會想把這當做秘密

Letting numbers stand for letters,

若將字母以數字代替

you can put the numbers in a matrix

你可將數字整理成矩陣

and then an encryption key in another.

再將加密金鑰設為另一個矩陣

Multiply them together

將他們相乘

and you've got a new encoded matrix.

就會得到一個受密碼保護的矩陣

The only way to decode the new matrix

將新矩陣解碼並閱讀秘密訊息

and read the message

的唯一方法

is to have the key,

就是擁有解碼金鑰

that second matrix.

那第二個矩陣

There's even a branch of mathematics

更有一個數學領域

that uses matrices constantly,

無時無刻使用矩陣

called Linear Algebra.

叫做線性代數

If you ever get a chance to study Linear Algebra,

如果你有機會學習線性代數的話

do it, it's pretty awesome.

抓緊機會，它超酷的

But just remember,

只要記得

once you know how to use matrices,

你一旦學會如何使用矩陣

you can do pretty much anything.

什麼也難不倒你