By R. Raman
Booklet by way of Raman, R.
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All blocks being square and of the same order, and exhibit the recursion for finding the Bi and r i' Hence show further that, if all matrices Pi' R i , Qi are commutative, and there are v 2 blocks in each matrix, then the determinant of the matrix on the left is b('Yv) where 'Yo = I, 'Y1 = Pi' 'Yi == Pi'Y i- 1 Ri-IQi-1'Yi-2· 49. ' p) by the recursion 1po 1pI 1pi == 1, == A, == A1pi-1 - p21pi_2' + 1)O/sin20; and show that if A = 2 p cos 20, then 1p,! hence that if K = J = V p sin2(v + JT, its proper values are Av(K) = 2 cos 20 v, 20 v == 2vO I == v1T/(n + 1) (this matrix arises repeatedly in the numerical solution of differential equations), PROBLEMS AND EXERCISES 35 50.
If A in Exercise 3 is normal, then T is diagonal; in particular, if A is Hermitian, T is also real, or if A is unitary, then I T I = I. 6. If T is upper triangular, and Tii i=- "'ij' then a can be chosen so that for j >i eiTE(e i , ej ; -a)TE(e i , ej ; a)e j = O. ), each Ri being upper (lower) triangular and having all diagonal elements equal. 7. , similar to a normal matrix). 8. A nilpotent matrix can have only 0 as a proper value; any proper value of an idempotent matrix is either 0 or a root of unity.
But this implies that which is the last of (2). An analogous argument leads to the inequalities on the left. As a particular case, K > 0 => II X 112 == K II X lub 1 (A) Ill' == lub 2 (A). This theorem implies that all norms are equivalent, in the sense that if the sequence II Xl - x 11K' 1 II x 2 -- x II K 1 ' • •• vanishes in the limit, then also the sequence X 11K' 2 vanishes in the limit, and in either event the sequence of vectors Xi can be said to have the limit x. Analogous remarks apply to matrices.