How to organize, add and multiply matrices - Bill Shillito

如何对矩阵进行组织，相加以及相乘 - 比尔 - 希利托

Translator: Andrea McDonough Reviewer: Bedirhan Cinar

翻译人员: xuanyu shi 校对人员: Erli Cai

By now, I'm sure you know

目前为止，我确定你知道你在生活中做的所有事都需要数字。

that in just about anything you do in life,

you need numbers.

In particular, though,

尽管，尤其在一些领域中不只是一些，

some fields don't just need a few numbers,

they need lots of them.

而是很多。

你是如何记录这些数字的呢？

How do you keep track of all those numbers?

Well, mathematicians dating back

好吧，早在中国古代的数学家

as early as ancient China

想出了用数组的方法来一次性表示很多数字。

came up with a way to represent

arrays of many numbers at once.

如今，我们称这样的数组是“矩阵”

Nowadays we call such an array a "matrix,"

其中许多数组一起叫做“矩阵”。

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

矩阵无处不在。

Matrices are everywhere.

They are all around us,

它们在我们的身边，

even now in this very room.

甚至在这个很小的房间里。。。

Sorry, let's get back on track.

抱歉，还是让我们回到主题上来。

Matrices really are everywhere, though.

矩阵真的是无处不在，但主要的,它们被运用在商业，经济学,密码学,物理,电子，

They are used in business,

economics,

cryptography,

physics,

electronics,

and computer graphics.

以及计算机图形。

One reason matrices are so cool

矩阵很棒的其中一个原因就是，

is that we can pack so much information into them

我们可以把很多的信息放进去，

and then turn a huge series of different problems

然后将一系列不同的问题

into one single problem.

变成一个单独的问题。

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

因此，为了可以使用矩阵， 我们需要知道它们的工作原理。

事实证明，你可以把矩阵

It turns out, you can treat matrices

就像普通的数字一样看待。

just like regular numbers.

You can add them,

你可以将它们相加,相减,甚至是相乘。

subtract them,

even multiply them.

You can't divide them,

你不能将它们相除，

but that's a rabbit hole of its own.

但是那可以通过其他方法实现。

Adding matrices is pretty simple.

矩阵相加是很简单的。

All you have to do is add the corresponding entries

你需要做的就是

将相应的项按照他们的顺序相加

in the order they come.

So the first entries get added together,

所以，第一项对应相加，

the second entries,

然后是第二项，

the third,

all the way down.

然后是第三项，

以此类推。

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

当然，你的矩阵必须具有相同的尺寸，

but that's pretty intuitive anyway.

但这只是非常直观而已。

You can also multiply the whole matrix

你也可以用整个矩阵和一个数字相乘，

by a number, called a scalar.

这个数字称为标量。

Just multiply every entry by that number.

只需要把矩阵里的每一项都乘以那个数字。

等下，还有更精彩的！

But wait, there's more!

You can actually multiply one matrix by another matrix.

事实上你可以令一个矩阵和另一个矩阵相乘。

然而，这个和它们一项一项相加是不一样的。

It's not like adding them, though,

where you do it entry by entry.

It's more unique

它更加特别一点,

and pretty cool once you get the hang of it.

一旦你得到了它的窍门是很棒的。

Here's how it works.

下面就是它怎么工作的。

Let's say you have two matrices.

比方说，你有两个矩阵。

我们把它们两个两个分，

Let's make them both two by two,

meaning two rows by two columns.

也就是说有两行两列。

把第一个矩阵写在左边，

Write the first matrix to the left

and the second matrix goes next to it

第二个矩阵写在第一个矩阵右边，

and translated up a bit,

然后向上移一点，

kind of like we are making a table.

有点像我们制作一个表格。

The product we get when we multiply the matrices together

我们得到的两个矩阵相乘的结果

will go right between them.

会在两个矩阵的中间位置。

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

我们还需要画一些网格线来帮助我们。

现在,来看第一个矩阵的第一行

Now, look at the first row of the first matrix

and the first column of the second matrix.

和第二个矩阵的第一列。

See how there's two numbers in each?

看到行和列的那两个数字了么?

Multiply the first number in the row

用第一行的第一个数字

乘以第一列的第一个数字：

by the first number in the column:

1 times 2 is 2.

1乘以2 是2。

Now do the next ones:

现在做剩下的，

3 times 3 is 9.

3乘以3是9。

Now add them up:

现在，将这两个数字相加：

2加9等于11。

2 plus 9 is 11.

现在我们把那个数字让在矩阵的左上方，

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

so that it matches up with the rows and columns

所以它和我们习惯的矩阵的行和列相匹配。

we used to get it.

See how that works?

看到它是如何做到的么？

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

你可以对其他的每一项做一样的事。

-4 plus 0 is -4.

负4 加上0 等于负4。

4 plus -3 is 1.

4加上负3等于1。

-8 plus 0 is -8.

负8加上0等于负8。

So, here's your answer.

所以,这是你的答案。

Not all that bad, is it?

并没有那么糟糕吧?

There's one catch, though.

不过,这里有一个陷阱。

Just like with addition,

就和加法一样，

your matrices have to be the right size.

你的矩阵必须有相同的大小。

Look at these two matrices.

看看这两个矩阵。

2 times 8 is 16.

2乘以8是16。

3 times 4 is 12.

3乘以4是12。

3 times

3乘以....

等一下。

wait a minute,

there are no more rows in the second matrix.

第二个矩阵中少了一行。

We ran out of room.

我们没有地方了继续做下去了。

So, these matrices can't be multiplied.

所以，这些矩阵不可以相乘。

The number of columns in the first matrix

第一个矩阵的列的数目

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

必须和第二个矩阵的行数相同。

只要你小心地匹配矩阵的尺寸，

As long as you're careful

to match up your dimensions right, though,

it's pretty easy.

不过呢，这个是非常容易的。

Understanding matrix multiplication

顺便说一下，

is just the beginning, by the way.

理解矩阵的乘法运算才是刚刚开始。

There's so much you can do with them.

你还可以用矩阵做很多事。

For example, let's say you want

举个例子来说，

to encrypt a secret message.

假设你想要给一个秘密信息加密。

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

假设是用“数学规则”。

Though, why anybody would want to keep this a secret

尽管，我不知道为什么

is beyond me.

除了我之外的人都想要保守这个秘密。

Letting numbers stand for letters,

让数字代表的字母，

you can put the numbers in a matrix

你可以把这些数字放在一个矩阵里，

and then an encryption key in another.

然后用另一个矩阵作为加密钥匙。

Multiply them together

and you've got a new encoded matrix.

把这两个矩阵相乘，

那么你就得到一个新的加密的矩阵。

The only way to decode the new matrix

解开新矩阵密码来

and read the message

is to have the key,

读取消息的唯一途径

必须有钥匙，

that second matrix.

那就是第二个矩阵。

There's even a branch of mathematics

甚至还有一个叫线性代数的数学分支经常使用矩阵。

that uses matrices constantly,

called Linear Algebra.

If you ever get a chance to study Linear Algebra,

如果你有机会学习线性代数,

do it, it's pretty awesome.

那就去学吧, 这个真的很酷。

But just remember,

但是要记住,

once you know how to use matrices,

一旦你知道如何使用矩阵,

you can do pretty much anything.

你几乎可以做任何事情。