好像目前还没有这方面题目的总结。这几天连续看到四个问这类题目的人，今天在这里简单写一下。这里我们不介绍其它有关矩阵的知识，只介绍矩阵乘法和相关性质。
不要以为数学中的矩阵也是黑色屏幕上不断变化的绿色字符。在数学中，一个矩阵说穿了就是一个二维数组。一个n行m列的矩阵可以乘以一个m行p列的矩阵，得到的结果是一个n行p列的矩阵，其中的第i行第j列位置上的数等于前一个矩阵第i行上的m个数与后一个矩阵第j列上的m个数对应相乘后所有m个乘积的和。比如，下面的算式表示一个2行2列的矩阵乘以2行3列的矩阵，其结果是一个2行3列的矩阵。其中，结果的那个4等于2*2+0*1：
下面的算式则是一个1 x 3的矩阵乘以3 x 2的矩阵，得到一个1 x 2的矩阵：
矩阵乘法的两个重要性质：一，矩阵乘法不满足交换律；二，矩阵乘法满足结合律。为什么矩阵乘法不满足交换律呢？废话，交换过来后两个矩阵有可能根本不能相乘。为什么它又满足结合律呢？仔细想想你会发现这也是废话。假设你有三个矩阵A、B、C，那么(AB)C和A(BC)的结果的第i行第j列上的数都等于所有A(ik)*B(kl)*C(lj)的和（枚举所有的k和l）。
经典题目1 给定n个点，m个操作，构造O(m+n)的算法输出m个操作后各点的位置。操作有平移、缩放、翻转和旋转
这里的操作是对所有点同时进行的。其中翻转是以坐标轴为对称轴进行翻转（两种情况），旋转则以原点为中心。如果对每个点分别进行模拟，那么m个操作总共耗时O(mn)。利用矩阵乘法可以在O(m)的时间里把所有操作合并为一个矩阵，然后每个点与该矩阵相乘即可直接得出最终该点的位置，总共耗时O(m+n)。假设初始时某个点的坐标为x和y，下面5个矩阵可以分别对其进行平移、旋转、翻转和旋转操作。预先把所有m个操作所对应的矩阵全部乘起来，再乘以(x,y,1)，即可一步得出最终点的位置。
经典题目2 给定矩阵A，请快速计算出A^n（n个A相乘）的结果，输出的每个数都mod p。
由于矩阵乘法具有结合律，因此A^4 = A * A * A * A = (A*A) * (A*A) = A^2 * A^2。我们可以得到这样的结论：当n为偶数时，A^n = A^(n/2) * A^(n/2)；当n为奇数时，A^n = A^(n/2) * A^(n/2) * A （其中n/2取整）。这就告诉我们，计算A^n也可以使用二分快速求幂的方法。例如，为了算出A^25的值，我们只需要递归地计算出A^12、A^6、A^3的值即可。根据这里的一些结果，我们可以在计算过程中不断取模，避免高精度运算。
经典题目3 POJ3233 (感谢rmq)
题目大意：给定矩阵A，求A + A^2 + A^3 + … + A^k的结果（两个矩阵相加就是对应位置分别相加）。输出的数据mod m。k<=10^9。
这道题两次二分，相当经典。首先我们知道，A^i可以二分求出。然后我们需要对整个题目的数据规模k进行二分。比如，当k=6时，有：
A + A^2 + A^3 + A^4 + A^5 + A^6 =(A + A^2 + A^3) + A^3*(A + A^2 + A^3)
应用这个式子后，规模k减小了一半。我们二分求出A^3后再递归地计算A + A^2 + A^3，即可得到原问题的答案。
经典题目4 VOJ1049
题目大意：顺次给出m个置换，反复使用这m个置换对初始序列进行操作，问k次置换后的序列。m<=10, k<2^31。
首先将这m个置换“合并”起来（算出这m个置换的乘积），然后接下来我们需要执行这个置换k/m次（取整，若有余数则剩下几步模拟即可）。注意任意一个置换都可以表示成矩阵的形式。例如，将1 2 3 4置换为3 1 2 4，相当于下面的矩阵乘法：
置换k/m次就相当于在前面乘以k/m个这样的矩阵。我们可以二分计算出该矩阵的k/m次方，再乘以初始序列即可。做出来了别忙着高兴，得意之时就是你灭亡之日，别忘了最后可能还有几个置换需要模拟。
经典题目5 《算法艺术与信息学竞赛》207页（2.1代数方法和模型，[例题5]细菌，版次不同可能页码有偏差）
大家自己去看看吧，书上讲得很详细。解题方法和上一题类似，都是用矩阵来表示操作，然后二分求最终状态。
经典题目6 给定n和p，求第n个Fibonacci数mod p的值，n不超过2^31
根据前面的一些思路，现在我们需要构造一个2 x 2的矩阵，使得它乘以(a,b)得到的结果是(b,a+b)。每多乘一次这个矩阵，这两个数就会多迭代一次。那么，我们把这个2 x 2的矩阵自乘n次，再乘以(0,1)就可以得到第n个Fibonacci数了。不用多想，这个2 x 2的矩阵很容易构造出来：
经典题目7 VOJ1067
我们可以用上面的方法二分求出任何一个线性递推式的第n项，其对应矩阵的构造方法为：在右上角的(n-1)*(n-1)的小矩阵中的主对角线上填1，矩阵第n行填对应的系数，其它地方都填0。例如，我们可以用下面的矩阵乘法来二分计算f(n) = 4f(n-1) – 3f(n-2) + 2f(n-4)的第k项：
利用矩阵乘法求解线性递推关系的题目我能编出一卡车来。这里给出的例题是系数全为1的情况。
经典题目8 给定一个有向图，问从A点恰好走k步（允许重复经过边）到达B点的方案数mod p的值
把给定的图转为邻接矩阵，即A(i,j)=1当且仅当存在一条边i->j。令C=A*A，那么C(i,j)=ΣA(i,k)*A(k,j)，实际上就等于从点i到点j恰好经过2条边的路径数（枚举k为中转点）。类似地，C*A的第i行第j列就表示从i到j经过3条边的路径数。同理，如果要求经过k步的路径数，我们只需要二分求出A^k即可。
经典题目9 用1 x 2的多米诺骨牌填满M x N的矩形有多少种方案，M<=5，N<2^31，输出答案mod p的结果
我们以M=3为例进行讲解。假设我们把这个矩形横着放在电脑屏幕上，从右往左一列一列地进行填充。其中前n-2列已经填满了，第n-1列参差不齐。现在我们要做的事情是把第n-1列也填满，将状态转
重载双目运算符尽量放到类定义的外面，以避免不同编译器的区别。
回复：谢谢指点；这方面的东西我还不太熟，过几天来研究
另外，matrix67大牛为什么不写一个矩阵快速乘法呢？这个一直弄不明白怎么做。
回复：是不是这个？
http://www.comp.nus.edu.sg/~xujia/mirror/algorithm.myrice.com/algorithm/commonalg/misc/strassen/strassen.htm
第6题的那个用矩阵乘法求第n个Fibonacci数的算法很感兴趣，可看的还是不大明白，能说的再详细一点吗？
回复：usera.imagecave.com/matrix67/fib_using_matrix.GIF
矩阵=matrix
经典题目9 能否再解释下
看不是很懂
左边的状态是怎么转移到右边的
111又是怎么转移到011的？
给个建议, 规范 C++ 书写.
#include <cstdio>, #define, scanf, printf 之类的东西不该出现哦.
第九题 超经典
能否写个pascal???
啊，太荣幸了，大牛居然贴我blog上的一个小题目 ~~
太开心了
回复：您牛的blog我每天都上，都是经典题目啊
您能告诉我您的博客网址吗？
楼主能把下面这个数列用矩阵的方式求出第n项的值吗？
数列用递归方式定义：
Q(1)= Q(2) =1
Q(N)=Q(N-Q(N-1))+Q(N-Q(N-2)) (N>2)
这个是非线性的啊
楼上的数列好象在VIJOS上看到过
其实类的双目运算符好像可以在类里面定义成友元函数……和在类外定义是等效的……这样看起来可能会比较舒服……
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最后一题能再讲得详细的吗？完全不理解。
“
以AT结尾、以AA结尾、以GG结尾、以?A结尾、以?G结尾、以?C结尾和以??结尾
”
是怎么得出的？
最后一题真的不好理解……
若能把所有状态之间的关系（即那个矩阵）列出来就好了
请问第4个题目中是对每个置换转换成n*n的矩阵，然后对每个矩阵相乘吗？如果是这样，当n=100时，时间复杂度也会达到n^3,最少也会是7^(logn),不会超时吗？
题目3
构造一个矩阵
A I
O I
自乘后
A^2 I+A
O I
乘3次为
A^3 I+A+A^2
O I
这样应该会更快吧，直接二分一下。
第九题能不能给我一个Pascal程序，我看不懂C++
对于题目2和3，只是1维情况的推广而已，对于2，可以用逐次平方计算（其实和m67给的递归方式差不了多少）
对于3，我能给出更好的方法，和问题2其实是类似的，基于二进制模式的递推（对应的递归是非对称的拆分），能在O(log(n))的时间给出解
http://www.earthson.net/archives/280
我笔拙，写得可能不是很好，但这不能掩盖这个算法的出色构造
我还给出了更一般的二进制可递推序列的特性要求和算法
http://www.earthson.net/archives/491
几个月前就看这篇了，今天重新学习，转载了一下并膜拜楼主一下
很详细表示很好理解~~
第九题真的很orz您的代码啊~
开始不明白const int validSet[]={0,3,6,12,15,24,27,30};的含义，琢磨明白之后只有orz的份了~
对于第三题这样子的可以这样做：这里的A就视为一个常量吧，k作为参数，那么这个式子就可以表示为f(k)，可以知道f(k)=f(k-1)*A+A，这个东西就可以化归到题目7中去了。如果这里面的A是矩阵的话，该用1的地方用单位矩阵就好了
//Complex(1);
Rot(x,thet){x*(sin(thet)+i*cos(thet));}
>> Rot(x, thet) #
x=Rot(x,2);
>> x*sin^2(2)-x*cos^2(2) + 2*x*sin(2)*cos(2)*i #
x=Rot(x,2);
>> x*sin^3(2)-3*x*sin(2)*cos^2(2) + (3*x*sin^2(2)*cos(2)-x*cos^3(2))*i #
Mov(x,a,b){x+a+b*i;}
>> Mov(x, a, b) #
Fx(x){-Conjugate(x);}
>> Fx(x) #
Fy(x){Conjugate(x);}
>> Fy(x) #
Mult(x,n){x*n;}
>> Mult(x, n) #
x=Rot(x,-2);
>> x*sin^3(2)*sin(-2)-3*x*sin^2(2)*cos(2)*cos(-2)-3*x*sin(2)*cos^2(2)*sin(-2)+x*cos^3(2)*cos(-2) + (x*sin^3(2)*cos(-2)+3*x*sin^2(2)*cos(2)*sin(-2)-3*x*sin(2)*cos^2(2)*cos(-2)-x*cos^3(2)*sin(-2))*i #
x=Mov(x,3,4);
>> x*sin^3(2)*sin(-2)-3*x*sin^2(2)*cos(2)*cos(-2)-3*x*sin(2)*cos^2(2)*sin(-2)+x*cos^3(2)*cos(-2)+3 + (x*sin^3(2)*cos(-2)+3*x*sin^2(2)*cos(2)*sin(-2)-3*x*sin(2)*cos^2(2)*cos(-2)-x*cos^3(2)*sin(-2)+4)*i #
Float(x);
>> 计算结果显示浮点数 #
Subst(x, x, 1);
>> sin^3(2)*sin(-2)-3*sin^2(2)*cos(2)*cos(-2)-3*sin(2)*cos^2(2)*sin(-2)+cos^3(2)*cos(-2)+3 + (sin^3(2)*cos(-2)+3*sin^2(2)*cos(2)*sin(-2)-3*sin(2)*cos^2(2)*cos(-2)-cos^3(2)*sin(-2)+4)*i #
for(i=0;i<n;i++)b[n]=a[a[n]];
好像可以在类里面定义成友元函数……和在类外定义是等效的……这样看起来可能会比较舒服……
最后少了一段T.T 应该是博客升级更新的时候丢了部分数据。。
百度了下补上
我们以M=3为例进行讲解。假设我们把这个矩形横着放在电脑屏幕上，从右往左一列一列地进行填充。其中前n-2列已经填满了，第n-1列参差不齐。现在我 们要做的事情是把第n-1列也填满，将状态转移到第n列上去。由于第n-1列的状态不一样（有8种不同的状态），因此我们需要分情况进行讨论。在图中，我 把转移前8种不同的状态放在左边，转移后8种不同的状态放在右边，左边的某种状态可以转移到右边的某种状态就在它们之间连一根线。注意为了保证方案不重 复，状态转移时我们不允许在第n-1列竖着放一个多米诺骨牌（例如左边第2种状态不能转移到右边第4种状态），否则这将与另一种转移前的状态重复。把这8 种状态的转移关系画成一个有向图，那么问题就变成了这样：从状态111出发，恰好经过n步回到这个状态有多少种方案。比如，n=2时有3种方 案，111->011->111、111->110->111和111->000->111，这与用多米诺骨牌覆盖 3×2矩形的方案一一对应。这样这个题目就转化为了我们前面的例题8
啊。。貌似后面直接少了一个题。。 大牛有时间补上吧。
还可以用矩阵来解决soduko
Thanks!
T4可以做到O(m)（所以说m<=10真是太小了啊开到10^8吧（雾））
how?
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