A based federer spectral sequence and the rational homotopy by Smith S.B.

Posted by

By Smith S.B.

Best mathematics books

Additional info for A based federer spectral sequence and the rational homotopy of function spaces

Sample text

Thus the ij-entry of BT AT is b1j ; b2j ; . . ; bmj ai1 ; ai2 ; . . ; aim T  b1j ai1  b2j ai2  . .  bmj aim Thus ABT  BT AT , since the corresponding entries are equal. 14. Find the diagonal and trace 2 3 of each matrix: 2 1 3 6 2 (a) A  4 2 À5 8 5, (b) B  4 3 4 À2 9 À5 (a) 4 À7 0 3 8 9 5, 2 C 1 2 4 À5  À3 . 6 The diagonal of A consists of the elements from the upper left corner of A to the lower right corner of A or, in other words, the elements a11 , a22 , a33 . Thus the diagonal of A consists of the numbers 1; À5, and 9.

Namely, trA  a11  a22  a33  . .  ann The following theorem applies. 7. Suppose A  aij  and B  bij  are n-square matrices and k is a scalar. Then: trAT   trA, (i) trA  B  trA  trB, (iii) (ii) (iv) trAB  trBA. 6. 4, trAT   trA; trA  B  trA  trB; tr2A  2 trA Furthermore, although AB T BA, the traces are equal. Identity Matrix, Scalar Matrices The n-square identity or unit matrix, denoted by In, or simply I, is the n-square matrix with 1's on the diagonal and 0's elsewhere.

Then A2  , which is diagonal. 24. Find an upper triangular matrix A such that A  0 3  À57 . 27   x y Set A  . Then x3  8, so x  2; and z3  27, so z  3. Next calculate A3 using x  2 and 0 z y  3:           2 y 2 y 2 y 4 5y 4 5y 8 19y   A2  and A3  0 3 0 3 0 3 0 9 0 9 0 27   2 À3 Thus 19y  À57, or y  À3. Accordingly, A  . 25. Let A  aij  and B  bij  be upper triangular matrices. Prove that AB is upper triangular with diagonal a11 b11 , a22 b22 ; . . ; ann bnn .