Homework4_11

Homework4_11 - 3 .-1 . .-1 . 3 .-2 . .-2 . 6 . . To avoid...

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Math Biophysics, Fall 2011 Homework 4 (1) Suppose G and M are real matrices with G = M 0 M . (a) Show that all the eigenvalues of G are non-negative. (b) If M is a 3 × n matrix show there can be at most 3 non-zero eigen- vectors. (Hint: What is the dimension of the image of the linear transformation M ? Same question for G .) (2) Suppose M and N are 3 × 3 matrices whose columns are linearly indepen- dent vectors. Suppose the gram matrices M 0 M and N 0 N are equal. Show that there is an orthogonal matrix S such that M = SN . (3) We say that A is a square root of G if A 0 A = G . Using Maple and the technique described in the notes, find a square root of
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Unformatted text preview: 3 .-1 . .-1 . 3 .-2 . .-2 . 6 . . To avoid complicated expressions, do this problem numerically. (4) Using Maple, nd coordinates of points in 3D space having the distance matrix 0 4 4 8 3 4 0 8 4 3 4 8 0 4 3 8 4 4 0 3 3 3 3 3 0 . You can use commands from the worksheet DistMatrix4 on the website. Give the answer to 2 decimal places. 1...
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