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,6,Click to edit Master title style,1,Click to edit Master text styles,Second level,Third level,线性变换,线性变换,Hopfield,网络的问题,网络的输出重复地乘以权矩阵,W。,这个重复操作的结果是什么？,输出是收敛,趋向无穷大，振荡？,这一章将研究矩阵乘法，它表示了一般的线性变换。,Hopfield 网络的问题网络的输出重复地乘以权矩阵 W。,线性变换,一个变换包含三个部分：,1.,一个被称为定义域的元素集合,X,=,x,i,，,2.,一个被称为定义域的元素集合,Y,=,y,i,，,和,3.,一个将每个,x,i,X,和一个元素,y,i,Y,相联系的规则。,一个变换是线性的，如果：,1.,对所有的,x,1,，,x,2,X,，,A,(,x,1,+,x,2,)=,A,(,x,1,)+,A,(,x,2,)，,2.,对所有的,x,X,，,a,，,A,(,a,x,)=,a,A,(,x,)。,线性变换一个变换包含三个部分：一个变换是线性的，如果：,例子,旋转变换,旋转变换是否是线性变换,?,1.,2.,例子 旋转变换旋转变换是否是线性变换?1.2.,矩阵表示,-(1),两个有限维向量空间之间的任意线性变换都可由矩阵的乘法来表示。,设,v,1,v,2,.,v,n,是向量空间,X,的一个基，,u,1,u,2,.,u,m,是向量空间,Y,的一个基。即对任意两个向量,x,X,和,y,Y,，有,设,A,:,X,Y,矩阵表示-(1)两个有限维向量空间之间的任意线性变换都可,矩阵表示,-(2),因为,A,是一个线性运算，,因为,u,i,是,Y,的基集,(The coefficients,a,ij,will make up the matrix representation of the transformation.),矩阵表示-(2)因为 A 是一个线性运算，因为 ui,矩阵表示,-(3),由于,u,i,是线性无关的,a,11,a,12,a,1,n,a,21,a,22,a,2,n,a,m,1,a,m,2,a,m,n,x,1,x,2,x,n,y,1,y,2,y,m,=,这等价于矩阵乘法,矩阵表示-(3)由于 ui 是线性无关的,a11a12,小结,A linear transformation can be represented by matrix multiplication.,To find the matrix which represents the transformation we must transform each basis vector for the domain and then expand the result in terms of the basis vectors of the range.,Each of these equations gives us,one column of the matrix.,小结A linear transformation can,例子,-(1),Stand a deck of playing cards on edge so that you are looking at the deck sideways.Draw a vector,x,on the edge of the deck.Now“skew”the deck by an angle,q,as shown below,and note the new vector,y,=,A,(,x,).What is the matrix of this transforma-tion in terms of the standard basis set?,例子-(1)Stand a deck of playin,例子,-(2),To find the matrix we need to transform each of the basis vectors.,We will use the standard basis vectors for both the domain and the range.,例子-(2)To find the matrix we,例子,-(3),We begin with,s,1,:,This gives us the first column of the matrix.,If we draw a line on the bottom card and then skew the,deck,the line will not change.,例子-(3)We begin with s1:This,例子,-(4),Next,we skew,s,2,:,This gives us the second column of the matrix.,例子-(4)Next,we skew s2:This,例子,-(5),变换矩阵是：,例子-(5)变换矩阵是：,基变换,Consider the linear transformation,A,:,X,Y,.Let,v,1,v,2,.,v,n,be a basis for,X,and let,u,1,u,2,.,u,m,be a basis for,Y,.,The matrix representation is:,a,11,a,12,a,1,n,a,21,a,22,a,2,n,a,m,1,a,m,2,a,m,n,x,1,x,2,x,n,y,1,y,2,y,m,=,基变换Consider the linear transfo,新基集,Now lets consider different basis sets.Let,t,1,t,2,.,t,n,be a basis for,X,and let,w,1,w,2,.,w,m,be a basis for,Y,.,The new matrix representation is:,a,11,a,12,a,1,n,a,21,a,22,a,2,n,a,m,1,a,m,2,a,m,n,x,1,x,2,x,n,y,1,y,2,y,m,=,新基集Now lets consider differen,A,和,A,之间的关系,?,Expand,t,i,in terms of the original basis vectors for,X,.,Expand,w,i,in terms of the original basis vectors for,Y,.,w,i,w,1,i,w,2,i,w,m,i,=,t,i,t,1,i,t,2,i,t,n,i,=,A 和 A 之间的关系?Expand ti in term,A,和,A,之间的关系,?(,续),B,t,t,1,t,2,t,n,=,相似变换,A 和 A 之间的关系?(续)Btt1t2tn=相似变,例子,-(1),Take the skewing problem described previously,and find the,new matrix representation using the basis set,s,1,s,2,.,t,1,0.5,1,=,t,2,1,1,=,(Same basis for,domain and range.),例子-(1)Take the skewing probl,例子,-(2),For,q,=45:,例子-(2)For q=45:,例子,-(3),Try a test vector:,Check using reciprocal basis vectors:,例子-(3)Try a test vector:Chec,特征值和特征向量,Let,A,:,X,X,be a linear transformation.Those vectors,z,X,which are not equal to zero,and those scalars,l,which satisfy,A,(,z,)=,l,z,are called eigenvectors and eigenvalues,respectively.,Can you find an eigenvector,for this transformation?,特征值和特征向量Let A:XX be a linear,计算特征值,Skewing example(45):,对于这个变换只有一个特征向量。,21,0,=,z,0,1,0,0,z,1,0,1,0,0,z,11,z,21,0,0,=,=,计算特征值Skewing example(45):对于这,对角化,Perform a change of basis(similarity transformation)using,the eigenvectors as the basis vectors.If the eigenvalues are,distinct,the new matrix will be diagonal.,Eigenvectors,Eigenvalues,n,B,1,A,B,l,1,0,0,0,l,2,0,0,0,l,=,对角化Perform a change of basis(,例子,对角形式：,例子对角形式：,
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