神經(jīng)網(wǎng)絡(luò)-配套-Ch6pres課件

1、線性變換線性變換Hopfield 網(wǎng)絡(luò)的問題網(wǎng)絡(luò)的輸出重復(fù)地乘以權(quán)矩陣 W。這個(gè)重復(fù)操作的結(jié)果是什么？輸出是收斂, 趨向無窮大，振蕩？這一章將研究矩陣乘法，它表示了一般的線性變換。Hopfield 網(wǎng)絡(luò)的問題網(wǎng)絡(luò)的輸出重復(fù)地乘以權(quán)矩陣 W。線性變換一個(gè)變換包含三個(gè)部分：1.一個(gè)被稱為定義域的元素集合 X = xi，2.一個(gè)被稱為定義域的元素集合 Y = yi，和3.一個(gè)將每個(gè) x i X 和一個(gè)元素 yi Y 相聯(lián)系的規(guī)則。一個(gè)變換是線性的，如果：1.對(duì)所有的 x 1，x 2 X，A(x 1 + x 2 ) = A(x 1) + A(x 2 )，2. 對(duì)所有的 x X，a ，A(a x ) =

2、 a A(x ) 。線性變換一個(gè)變換包含三個(gè)部分：一個(gè)變換是線性的，如果：例子 旋轉(zhuǎn)變換旋轉(zhuǎn)變換是否是線性變換?1.2.例子 旋轉(zhuǎn)變換旋轉(zhuǎn)變換是否是線性變換?1.2.矩陣表示 - (1)兩個(gè)有限維向量空間之間的任意線性變換都可由矩陣的乘法來表示。設(shè) v1, v2, ., vn 是向量空間 X 的一個(gè)基，u1, u2, ., um 是向量空間 Y 的一個(gè)基。即對(duì)任意兩個(gè)向量 x X 和 y Y ，有設(shè) A:XY矩陣表示 - (1)兩個(gè)有限維向量空間之間的任意線性變換都可矩陣表示 - (2)因?yàn)?A 是一個(gè)線性運(yùn)算， 因?yàn)?ui 是 Y 的基集,(The coefficients aij will

3、 make up the matrix representation of the transformation.)矩陣表示 - (2)因?yàn)?A 是一個(gè)線性運(yùn)算， 因?yàn)?ui 矩陣表示 - (3)由于 ui 是線性無關(guān)的,a11a12a1na21a22a2nam1am2amnx1x2xny1y2ym=這等價(jià)于矩陣乘法矩陣表示 - (3)由于 ui 是線性無關(guān)的,a11a12小結(jié)A linear transformation can be represented by matrix multiplication.To find the matrix which represents the tr

4、ansformation 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.小結(jié)A linear transformation can 例子 - (1)Stand a deck of playing cards on edge so that you are looking at the d

5、eck 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

6、 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 s1:This gives us the first column of the matrix.If we draw a line on the bottom card and then skew thedeck, the line will not change.例子 - (3)We begin wit

7、h s1:This 例子 - (4)Next, we skew s2:This gives us the second column of the matrix.例子 - (4)Next, we skew s2:This 例子 - (5)變換矩陣是：例子 - (5)變換矩陣是：基變換Consider the linear transformation A:XY. Let v1, v2, ., vn be a basis for X, and let u1, u2, ., um be a basis for Y. The matrix representation is:a11a12a1na21

8、a22a2nam1am2amnx1x2xny1y2ym=基變換Consider the linear transfo新基集Now lets consider different basis sets. Let t1, t2, ., tn be a basis for X, and let w1, w2, ., wm be a basis for Y. The new matrix representation is:a11a12a1na21a22a2nam1am2amnx1x2xny1y2ym=新基集Now lets consider differenA 和 A 之間的關(guān)系?Expand ti

9、 in terms of the original basis vectors for X.Expand wi in terms of the original basis vectors for Y.wiw1iw2iwmi=tit1it2itni=A 和 A 之間的關(guān)系?Expand ti in termA 和 A 之間的關(guān)系? (續(xù))Btt1t2tn=相似變換A 和 A 之間的關(guān)系? (續(xù))Btt1t2tn=相似變例子 - (1)Take the skewing problem described previously, and find thenew matrix representat

10、ion using the basis set s1, s2.t10.51=t211=(Same basis fordomain 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:XX be a linear transformation. Those vectorsz X,

11、which are not equal to zero, and those scalarsl which satisfy A(z) = l zare called eigenvectors and eigenvalues, respectively.Can you find an eigenvectorfor this transformation?特征值和特征向量Let A:XX be a linear 計(jì)算特征值Skewing example (45):對(duì)于這個(gè)變換只有一個(gè)特征向量。210=z0100z10100z11z2100=計(jì)算特征值Skewing example (45):對(duì)于這對(duì)角化Perform a change of basis (similarity transformation) usingthe eigenv