Key to MATLAB Exercise 4 School of Mathematical Sciences Xiamen Univer http:/edjpkc.xmu.ed 0.2427-0.40ll >>A=[01;10}[X1,D=eig(A) -0.70710.7071 0.70710.7071 > rank(X1size(X1, 1) ans The answer is Xl= 0707107071//-10 -0.70710.7071 Here Xl is an invertible matrix We may check Xl by other functions such as det, rref and so on [X2, -0.70710.7071 ans 0 Therefore X2 is the answer too 0.50000.5000 0.50000.5000 0 ans Therefore X3 is also the answer The keys to 2)to 6)are omittedKey to MATLAB Exercise 4  School of Mathematical Sciences Xiamen University  http://gdjpkc.xmu.edu.cn  Key to Ex4­3  ­0.2427 ­ 0.4011i.  4． 1) >> A=[0 1;1 0]; [X1, D]=eig(A) X1= ­0.7071  0.7071  0.7071  0.7071  D = ­1  0  0  1  >> rank(X1)==size(X1, 1) ans =  1  The answer is 0.7071 0.7071 1 0.7071 0.7071 X Ê - ˆ = Á ˜ Ë ¯ ,  1 0  0 1 D Ê- ˆ = Á ˜ Ë ¯ . Here X1 is an invertible matrix.  We may check X1 by other functions such as det, rref and so on.  Or >> [X2, S]=schur(A) X2 = ­0.7071  0.7071  0.7071  0.7071  S =  ­1  0  0  1  >> X2*S*inv(X2) ans =  0  1.0000  1.0000  0  Therefore X2 is the answer, too.  Or >> [X3,J]=jordan(A) X3 = 0.5000  0.5000  ­0.5000  0.5000  J =  ­1  0  0  1  >> X3*J*inv(X3) ans =  0  1  1  0  Therefore X3 is also the answer.  The keys to 2) to 6) are omitted