Let n be a positive integer and let A=[a_(ij)]_(n imes n) be the n imes n matrix with a_(ij)={(2,i=j,),(-1,j=i+-1, for ),(0, otherwise ):}
all i,j=1,2,dots,n, that is,
A=[[2,-1,0,0,cdots,0,0,0],[-1,2,-1,0,cdots,0,0,0],[0,-1,2,-1,cdots,0,0,0],[0,0,-1,2,cdots,0,0,0],[vdots,vdots,vdots,vdots,ddots,vdots,vdots,vdots],[0,0,0,0,cdots,2,-1,0],[0,0,0,0,cdots,-1,2,-1],[0,0,0,0,cdots,0,-1,2]].
Prove that every eigenvalues of A is a positive real number.
2,
i=j, j=i1, for otherwise
8. Let n be a positive integer and let A= [ai],x. be the n n matrix with aii
lo,
all i, j=1, 2,..., n, that is,
2 1 0 1 2 1 0 - 1 2 0 0 1 . ... ... 0 0 0 0 0 0 0 0 0
0 0 ... 1 ... 2 ... ... *.. 0 - 0 0
0 0 0 0 ... 2 1 0
0 0 0 0 ... -1 2 1
0] 0 0 0 ... 0
A=
2 ]
Prove that every eigenvalues of A is a positive real number.
