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### hw11problems

Course: MATH 245, Fall 2009
School: W. Alabama
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Word Count: 399

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245, MATH Fall 2008 Assignment 11 Not to be handed in. 1. Show that every normal matrix in M22 (R) is either of the form a -b b a or a b b c 2. Let Q Mnn (R) be an orthogonal matrix. If tr(Q) &gt; n - 2 prove that det Q = 1. 3. Let A, B Mnn (C) be complex symmetric matrices, i.e. A = At , B = B t . Prove that A is congruent to B if and only if A and B have the same rank. 4. Let Skewn be the vector space of...

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245, MATH Fall 2008 Assignment 11 Not to be handed in. 1. Show that every normal matrix in M22 (R) is either of the form a -b b a or a b b c 2. Let Q Mnn (R) be an orthogonal matrix. If tr(Q) > n - 2 prove that det Q = 1. 3. Let A, B Mnn (C) be complex symmetric matrices, i.e. A = At , B = B t . Prove that A is congruent to B if and only if A and B have the same rank. 4. Let Skewn be the vector space of all real skew-symmetric n n matrices. (a) Show that for any matrix A Skewn , there is an orthogonal matrix Q and positive real numbers 1 , . . . , k (k n/2) such that 1 J 2 J .. . 0 1 , k J where J = ,. Qt AQ = -1 0 0 .. . 0 (b) Show that there is a linear transformation : 2 (Rn ) Skewn , such that (x y) = col xy t - yxt , for all x, y Rn . Show that is an isomorphism. col (c) Show that for any matrix Q Mnn (R), ((LQ ) ()) = Q()Qt . Using part (a), deduce that any vector 2 (Rn ) can be written in the form k = i=1 i xi yi , with k n/2. (d) Suppose n is even. Let A = () for standard basis for Rn . Since dim n (Rn = ) 1, 2 (Rn ), and let {e1 , . . . , en } denote the 1 = Pf(A) e1 e2 en . (n/2)! n/2 where Pf(A) R is a scalar depending on A. Pf(A) is called the Pfaffian of the skew-symmetric matrix A. Prove that 1 2 k if 2k = rank(A) = n Pf(A) = 0 otherwise,, where 1 , . . . , k are the scalars from part (a). Deduce that Pf(A)2 = det(A). 5. The goal of following exercise is to give an alternate proof of Sylvester's criterion. Let A be an n n real symmetric matrix such that det(A(i) ) > 0 for i = 1, . . . , n, where A(i) is the i i submatrix of A in the upper left corner. (a) Let P = span{e1 , . . . , en-1 }, where {e1 ,...

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1.3250000 1.9250001 743.68880 220.61449 1.9250001 2.0750000 1777.0221 458.75330 2.0750000 2.4650000 850.98224 280.86813 2.4650000 3.2450000 596.09554
Berkeley - ASTRO - 00312047
1.3250000 1.9250001 743.68880 220.61449 1.9250001 2.0750000 1777.0221 458.75330 2.0750000 2.4650000 850.98224 280.86813 2.4650000 3.2450000 596.09554
Berkeley - ASTRO - 00312047
# Time [days] Mag Magerr Band Uplim Ref 0.00122 19.59 0.12 white no GCN7758 0.00123 19.8 0.1 White no GCN7753 0.00244 20.26 0.38 v no GCN7758 0.0
Berkeley - ASTRO - 00312047
26280.3 23 0.5 R no
Berkeley - ASTRO - 00312047
26280.3 2.375476e-06 1.093947e-06 R no
Berkeley - ASTRO - 00312047
1.35501 1 1
Berkeley - ASTRO - 00312047
1.8899999 207.26900 8.0948969 -1.6837656 18792.640 207.26900 167549.64 4.4988381 -1.0247551 0.76540139
Berkeley - ASTRO - 00312047
chi^2/nu= 19.788363 / 15The fit is rejectable at 81.979747 % Confidence 1.92500 2.07500 1196.7708 2.07500 2.46500 1136.9505 2.46500 3.24500 1055.5888 3.24500 3.5150
Berkeley - ASTRO - 00312047
&lt;html&gt;&lt;head&gt;&lt;title&gt;Your NED Search Results&lt;/title&gt;&lt;/head&gt;&lt;body background=&quot;/pics/NEDbgHelp.gif&quot; bgcolor=&quot;#FFFFFF&quot;&gt;&lt;center&gt;&lt;font size=6 color=&quot;#CC3333&quot;&gt;&lt;b&gt;N&lt;/b&gt;&lt;/font&gt;&lt;font size=4 color=&quot;#000000&quot;&gt;&lt;b&gt;ASA/IPAC&lt;/b&gt;&lt;/font&gt;&amp;nbsp;&lt;font size=6 color=&quot;#CC
Berkeley - ASTRO - 00312047
104.995 133.405 1.41231 0.397191108.332 166 0.955355 0.188914166 208.624 0.375968 0.124193208.624 266.292 0.28387 0.0917293266.292 296.38 0.500004 0.166672296.38 374.107 0.212089 0.0680407374.107 454.34 0.223431 0.0691595454.34 567.169 0.14100
Berkeley - ASTRO - 00312047
Source Contamination: 1.16E-05 +/- 1.1E-06 cts/s
Berkeley - ASTRO - 00312047
output00312047000_999/sw00312047000xpcw2po_cl.evtoutput00312047001_999/sw00312047001xpcw2po_cl.evtoutput00312047002_999/sw00312047002xpcw2po_cl.evtoutput00312047003_999/sw00312047003xpcw2po_cl.evtoutput00312047003_999/sw00312047003xpcw3po_cl.evt
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# t1 t2 dt rad_min rad_max cts err scl bg bg_rat wt 0.104995 0.106968 0.001973 0. 16. 2.00 1.41 0.900829 0.000000 0.426667 1 0.108332 0.120868 0.012536 0. 16. 10.86
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# tmin tmax 0.108332 167.551 [ksec];instrument XRT;exposure 15685.167;xunit kev;bintype counts0.000000 0.010000 0.000000 0.0000000.010000 0.020000 0.000000 0.0000000.020000 0.030000 0.000000 0.0000000.030000 0.040000 0.000000 0
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# tmin tmax 0.10499500 5.41227 [ksec];instrument XRT;exposure 3.9661232;xunit kev;bintype counts0.000000 0.010000 0.000000 0.0000000.010000 0.020000 0.000000 0.0000000.020000 0.030000 0.000000 0.0000000.030000 0.040000 0.00000
Berkeley - ASTRO - 00312047
output00312047000_999/sw00312047000xwtw2po_cl.evtoutput00312047001_999/sw00312047001xwtw2po_cl.evtoutput00312047002_999/sw00312047002xwtw2po_cl.evt
Berkeley - ASTRO - 00312047
SIMPLE = T / file does conform to FITS standardBITPIX = 8 / number of bits per data pixelNAXIS = 0 / number of data axesEXTEND = T / FITS dataset may contain extensio
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# Ep dEp lprob lEiso dlEiso11.084 0.009 -1.05e-05 117.520 0.12311.094 0.010 -2.47e-05 117.495 0.12911.105 0.012 -3.51e-05 117.495 0.12911.117 0.013 -3.49e-05 117.495 0.12911.132 0.015 -2.96e-05 117.495 0.12911.148 0.018 -1.40e-04 117.495 0.129
Berkeley - ASTRO - 00312047
# Ep lEiso1.002 120.0651.004 119.3301.017 121.5141.025 121.2391.029 120.3521.038 119.4311.044 121.7941.049 122.2891.057 119.5291.057 119.5661.063 120.4801.072 120.0511.076 121.3501.083 120.7281.090 120.1471.093 119.2561.106 120.5191
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# Ep dEp lprob lNiso dlNiso11.084 0.009 -1.05e-05 135.543 0.21111.094 0.010 -2.02e-05 135.493 0.22011.105 0.012 -3.43e-05 135.493 0.21911.117 0.013 -4.78e-05 135.493 0.21911.132 0.015 -1.23e-05 135.493 0.21911.148 0.018 -1.40e-04 135.493 0.219
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# Ep lNiso1.002 139.3421.004 138.6201.017 140.7591.025 140.4871.029 139.6131.038 138.1861.044 141.0291.049 141.5151.057 138.7931.057 138.8301.063 139.7291.072 139.3021.076 140.5841.083 139.9681.090 139.3911.093 140.8941.106 139.7541
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# tmin tmax 10.0000 167.55140 [ksec];instrument XRT;exposure 13182.346;xunit kev;bintype counts0.000000 0.010000 0.000000 0.0000000.010000 0.020000 0.000000 0.0000000.020000 0.030000 0.000000 0.0000000.030000 0.040000 0.00000
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