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

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Thus the ij-entry of BT AT is ‰b1j ; b2j ; . . ; bmj Š‰ai1 ; ai2 ; . . ; aim ŠT ˆ b1j ai1 ‡ b2j ai2 ‡ . . ‡ bmj aim Thus …AB†T ˆ 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 ˆ 4 2 À5 8 5, (b) B ˆ 4 3 4 À2 9 À5 (a) 4 À7 0 3 8 9 5, 2 Cˆ 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† ˆ a11 ‡ a22 ‡ a33 ‡ . . ‡ ann The following theorem applies. 7. Suppose A ˆ ‰aij Š and B ˆ ‰bij Š are n-square matrices and k is a scalar. Then: tr…AT † ˆ tr…A†, (i) tr…A ‡ B† ˆ tr…A† ‡ tr…B†, (iii) (ii) (iv) tr…AB† ˆ tr…BA†. 6. 4, tr…AT † ˆ tr…A†; tr…A ‡ B† ˆ tr…A† ‡ tr…B†; tr…2A† ˆ 2 tr…A† Furthermore, although AB Tˆ 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 ˆ , which is diagonal. 24. Find an upper triangular matrix A such that A ˆ 0 3  À57 . 27   x y Set A ˆ . Then x3 ˆ 8, so x ˆ 2; and z3 ˆ 27, so z ˆ 3. Next calculate A3 using x ˆ 2 and 0 z y ˆ 3:           2 y 2 y 2 y 4 5y 4 5y 8 19y ˆ ˆ A2 ˆ and A3 ˆ 0 3 0 3 0 3 0 9 0 9 0 27   2 À3 Thus 19y ˆ À57, or y ˆ À3. Accordingly, A ˆ . 25. Let A ˆ ‰aij Š and B ˆ ‰bij Š be upper triangular matrices. Prove that AB is upper triangular with diagonal a11 b11 , a22 b22 ; . . ; ann bnn .

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