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矩阵的组成和运算

How to organize, add and multiply matrices

现在 我相信你知道
By now, I’m sure you know
很多生活中的事
that in just about anything you do in life,
都需要依靠数字
you need numbers.
然而特别地
In particular, though,
一些领域不只需要单个的数
some fields don’t just need a few numbers,
他们需要很多数
they need lots of them.
怎么记录所有被需要的数字呢?
How do you keep track of all those numbers?
于是 早在古代中国
Well, mathematicians dating back
数学家们就约定了
as early as ancient China
使用一种方式来代表
came up with a way to represent
成组的数
arrays of many numbers at once.
于是现在我们叫这样的一个组为“矩阵”
Nowadays we call such an array a “matrix,”
许多的矩阵聚在一起 就是“matrices”(矩阵的复数)
and many of them hanging out together, “matrices”.
矩阵无处不在
Matrices are everywhere.
矩阵就在我们身边
They are all around us,
即使是现在这个情况
even now in this very room.
不好意思 让我们回到正轨
Sorry, let’s get back on track.
矩阵的确是无处不在的
Matrices really are everywhere, though.
他们被用于商业计算
They are used in business,
经济学
economics,
密码学
cryptography,
物理学
physics,
电子工业
electronics,
和计算机图形学
and computer graphics.
矩阵很厉害的其中一个理由是
One reason matrices are so cool
我们可以在矩阵中保存许多信息
is that we can pack so much information into them
然后将很复杂的问题相互联系
and then turn a huge series of different problems
转化为一个简单问题
into one single problem.
所以为了应用矩阵 我们需要知道他们如何运算
So, to use matrices, we need to learn how they work.
所以 你可以视矩阵为
It turns out, you can treat matrices
一个普通的数
just like regular numbers.
你可以使矩阵相加
You can add them,
相减
subtract them,
甚至相乘
even multiply them.
但他们不支持除法
You can’t divide them,
但这并没有什么关系
but that’s a rabbit hole of its own.
矩阵的加法非常简单
Adding matrices is pretty simple.
你需要做的事情只是对应位相加
All you have to do is add the corresponding entries
然后把他们放到原来的位置
in the order they come.
所以第一个元素被加起来了
So the first entries get added together,
第二个元素
the second entries,
第三个
the third,
矩阵被加起来了
all the way down.
当然 相加的矩阵的大小必须完全相同
Of course, your matrices have to be the same size,
不过这样描述有些不太严谨
but that’s pretty intuitive anyway.
你同样可以把矩阵
You can also multiply the whole matrix
和被称作标量的数字相乘
by a number, called a scalar.
仅仅是将矩阵的每个元素都乘一次这个数而已
Just multiply every entry by that number.
但是等等 还有更多
But wait, there’s more!
事实上你可以用一个矩阵乘另一个矩阵
You can actually multiply one matrix by another matrix.
这不像矩阵的加法了 虽然还是需要每个元素单独处理
It’s not like adding them, though, where you do it entry by entry.
矩阵间的乘法更加独特
It’s more unique
如果你理解了这种方式那真是太好了
and pretty cool once you get the hang of it.
让我们来看看它是如何进行的吧
Here’s how it works.
现在你有两个矩阵
Let’s say you have two matrices.
现在我们让矩阵大小都是2×2
Let’s make them both two by two,
意思是矩阵是两行两列的
meaning two rows by two columns.
把第一个矩阵写在左边
Write the first matrix to the left
第二个写在前一个旁边
and the second matrix goes next to it
然后把它向上移动一格
and translated up a bit,
像我们列表格一样
kind of like we are making a table.
当我们将两个矩阵相乘时得到新的结果矩阵
The product we get when we multiply the matrices together
就在它们的右下方
will go right between them.
我们画出一些网格来帮我们理解
We’ll also draw some gridlines to help us along.
现在 看第一个矩阵的第一行
Now, look at the first row of the first matrix
和第二个矩阵的第一列
and the first column of the second matrix.
看到行列上的各两个数字了么
See how there’s two numbers in each?
将行上的第一个数字
Multiply the first number in the row
和列上的第一个数字相乘
by the first number in the column:
1乘2等于2
1 times 2 is 2.
现在做下一个
Now do the next ones:
3乘3等于9
3 times 3 is 9.
现在把他们加起来
Now add them up:
2加9等于11
2 plus 9 is 11.
将数字放到左侧顶部的位置
Let’s put that number in the top-left position
所以这恰好和使用的行列位置
so that it matches up with the rows and columns
相匹配
we used to get it.
看看该如何计算呢
See how that works?
你可以用同样的方法求出另一个元素
You can do the same thing to get the other entries.
-4加0等于-4
-4 plus 0 is -4.
4加-3等于1
4 plus -3 is 1.
-8加0等于-8
-8 plus 0 is -8.
所以这就是答案了
So, here’s your answer.
不是那么难 对吧
Not all that bad, is it?
不过还有一点需要注意
There’s one catch, though.
像加法一样
Just like with addition,
你的矩阵大小必须相符合
your matrices have to be the right size.
看这两个矩阵
Look at these two matrices.
2乘8等于16
2 times 8 is 16.
3乘4等于12
3 times 4 is 12.
3乘……
3 times
等等
wait a minute,
第二个矩阵没有行了
there are no more rows in the second matrix.
我们在计算的时候超出了范围
We ran out of room.
所以 这样的矩阵不能相乘
So, these matrices can’t be multiplied.
第一个矩阵的列数
The number of columns in the first matrix
必须和第二个矩阵的行数相等
has to be the same as the number of rows in the second matrix.
所以要仔细
As long as you’re careful
要使矩阵的大小相符合
to match up your dimensions right, though,
尽管这非常容易
it’s pretty easy.
弄明白了矩阵的乘法
Understanding matrix multiplication
只是个开始 顺便一提
is just the beginning, by the way.
你可以用它做很多事情
There’s so much you can do with them.
例如 你现在想
For example, let’s say you want
加密一份机密消息
to encrypt a secret message.
消息是“Math rules”
Let’s say it’s “Math rules”.
然而 我们想要这个机密信息
Though, why anybody would want to keep this a secret
不被其它人知道
is beyond me.
把信息转换成数
Letting numbers stand for letters,
你可以把这些数放进矩阵中
you can put the numbers in a matrix
然后把密钥用同样的方法形成另一个矩阵
and then an encryption key in another.
将它们乘起来
Multiply them together
然后你就得到了一个新的加密矩阵
and you’ve got a new encoded matrix.
只有一个方式可以解码新矩阵
The only way to decode the new matrix
并得到信息
and read the message
那就是得到密钥
is to have the key,
那个第二矩阵
that second matrix.
实际上有一个数学分支
There’s even a branch of mathematics
会常常用到矩阵
that uses matrices constantly,
叫做线性代数
called Linear Algebra.
如果你有机会学习线性代数
If you ever get a chance to study Linear Algebra,
快去学吧 这会很有趣的
do it, it’s pretty awesome.
不过要记住
But just remember,
如果你知道怎么使用矩阵
once you know how to use matrices,
你可以做到更多事情
you can do pretty much anything.

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