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February 18, 2011

Two interesting proofs of the Cauchy-Schwarz inequality (complex case)

Filed under: mathematics — ckrao @ 10:06 am

In a previous post I gave two proofs of the Cauchy-Schwarz inequality for a real inner product space. Here I look at the case of a complex inner product space:

\displaystyle \left|\langle{x,y\rangle}\right| \leq \left\|x\right\| \left\|y\right\| \quad...(1)

One commonly used proof in the discrete case is via Lagrange’s identity. Here are two other nice proofs.

1. For any complex number \alpha we use the non-negativity of the square \displaystyle \left\|x-\alpha y\right\|^2:

\begin{array}{lcl} 0 &\leq & \left\|x-\alpha y\right\|^2\\&=& \left\|x|\right\|^2 + |\alpha|^2\left\|y\right\|^2 - \langle{x,\alpha y\rangle} - \langle{\alpha y,x\rangle}\\&=& \left(\begin{array}{cc} 1 & \overline{\alpha}\end{array}\right)\left( \begin{array}{cc} \left\|x\right\|^2 & \langle{x,y\rangle}\\ \langle{y,x\rangle} & \left\|y\right\|^2\end{array}\right)\left(\begin{array}{c} 1\\ \alpha \end{array}\right).\end{array}

Let \displaystyle A=\left( \begin{array}{cc} \left\|x\right\|^2 & \langle{x,y\rangle}\\ \langle{y,x\rangle} & \left\|y\right\|^2\end{array}\right). The above equation shows that if the 2 by 1 vector z = \left(\begin{array}{c} 1\\ \alpha \end{array}\right) has non-zero first component, then z^*A z \geq 0. This non-negativity also holds when z has zero first component as for \displaystyle z = \left(\begin{array}{c}0 \\ \alpha \end{array}\right) we have z^*A z = |\alpha|^2 \left\|y\right\|^2\geq 0. We conclude that A is positive semidefinite, so has non-negative determinant

\displaystyle 0 \leq \det A = \left\|x\right\|^2\left\|y\right\|^2 - \left|\langle{x,y\rangle} \right|^2,

from which (1) follows.

2. For convenience let \hat{y} = y/\left\|y\right\|. Consider the projection of x onto y, given by \langle{x,\hat{y}\rangle}\hat{y}. We compute the squared length of their difference.

\begin{array}{lcl} 0 & \leq & \left\|x- \langle{x,\hat{y}\rangle}\hat{y}\right\|^2\\&=& ||x||^2 + \left|\langle{x,\hat{y}\rangle} \right|^2 - 2|\langle{x,\hat{y}\rangle}|^2\\&=& \left\|x\right\|^2 - \left|\langle{x,\hat{y}\rangle}\right|^2 \\&=& \left\|x\right\|^2 - \frac{\left|\langle{x,y\rangle}\right|^2}{\left\|y\right\|^2},\end{array}

from which (1) easily follows. Equality holds iff the projection of x onto y is the zero vector: in other words, they are parallel or one of the vectors is zero.

Proof 2 shows that the Cauchy-Schwarz inequality is a consequence of Gram-Schmidt orthogonalisation applied to two vectors, also mentioned in [1].

Reference

[1] J. M. Steele, The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities, MAA Problem Books Series, 2004.

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February 13, 2011

Australia’s undefeated streak in the cricket World Cup

Filed under: cricket,sport — ckrao @ 9:55 am

With the 2011 cricket World Cup less than a week away, here  is a list (in chronological order) of the last 29 matches that Australia has played in the World Cups back to 1999. The last time Australia lost a game was to Pakistan in the 1999 World Cup. Update: table updated to include their first five games of the 2011 World Cup – the undefeated streak ended at 34 with a loss to Pakistan.

Opposition Result Aus Score Overs RPO Target Inn Aus batted in Notes
v Bangladesh won by 7 wickets with 181 balls to spare 181/3 19.5 9.12 179 2 at the time the highest score made in 20 overs
v West Indies won by 6 wickets with 56 balls to spare 111/4 40.4 2.72 111 2 deliberate go-slow tactics in an attempt to prevent New Zealand from progressing to the Super-6 stage, but they did
v India won by 77 runs 282/6 50 5.64 1 game decided in first 6.2 overs of 2nd innings when India were reduced to 4/17 with 3 wickets by McGrath
v Zimbabwe won by 44 runs 303/4 50 6.06 1 Neil Johnson’s 132* the highlight
v South Africa won by 5 wickets 272/5 49.4 5.47 272 2 Steve Waugh’s 120* saw them through to semi finals
v South Africa tied 213 49.2 4.31 1 one of the great ODI games, Warne superb with 4 wickets
v Pakistan
(1999 final)
won by 8 wickets with 179 balls to spare 133/2 20.1 6.59 133 2 Australia at their bowling and fielding best when it most counted
v Pakistan won by 82 runs 310/8 50 6.2 1 Andrew Symonds scored 143* off 125 coming in at 4/86 in the 16th
v India won by 9 wickets 128/1 22.2 5.73 126 2 India skittled for 125 with Gillespie picking up 3/13 off 10 overs
v Netherlands won by 75 runs 170/2 36 4.72 1 rain-affected game, in which Netherlands needed 198 from 36 overs
v Zimbabwe won by 7 wickets with 15 balls to spare 248/3 47.3 5.22 247 2 Australia never in doubt despite Zimbabwe posting a competitive total
v Namibia won by 256 runs 301/6 50 6.02 1 Highest winning margin in an ODI, Lehmann scoring 4, 4, 4, 6, 4 and 6 off the last over
v England won by 2 wickets with 2 balls to spare 208/8 49.4 4.18 205 2 Despite Bichel’s 7/20 to restrict England to 8/204, Caddick’s 4 top-order wickets had Australia reeling at 8/135 before Bevan and Bichel saw them through
v Sri Lanka won by 96 runs 319/5 50 6.38 1 Gilchrist run out for 99 and Ponting made 114. Aravinda de Silva (92 off 94) had no support with nobody else making more than 21
v New Zealand won by 96 runs 208/9 50 4.16 1 another dramatic match in which Australia recovered from 7/84 (Bond 6/23) to reach 208. Lee took 5 as NZ could only reply with 112.
v Kenya won by 5 wickets with 112 balls to spare 178/5 31.2 5.68 175 2 Aasif Karim had 3/7 off 8.2 overs with 6 maidens, but the result was never in doubt
v Sri Lanka won by 48 runs 212/7 50 4.24 1 Symonds’ knock of 91* crucial in taking Australia to 212 which proved more than enough with Lee’s 3 early wickets

v India
(2003 final)

won by 125 runs 359/2 50 7.18 1 Australia again saving its best for the final, with Ponting (140* off 121) and Martyn (88*) adding 234 for the 3rd wicket
v Scotland won by 203 runs 334/6 50 6.68 1 Australia scored 53 off the last 3 after Ponting’s 113 off 93
v Netherlands won by 229 runs 358/5 50 7.16 1 Hodge (123 off 89) and Clarke (93* off 85) added 204 in 164 balls
v South Africa won by 83 runs 377/6 50 7.54 1 Australia’s highest World Cup score set up by Hayden’s 101 off 68 followed by 90s by Ponting and Clarke. South Africa were 1/220 in the 32nd over before losing their last 9 wickets for 74.
v West Indies won by 103 runs 322/6 50 6.44 1 Australia’s 5th straight 300+ score in World Cup matches, set up by Hayden’s 158
v Bangladesh won by 10 wickets with 49 balls to spare 106/0 13.5 7.66 105 2 Rain-reduced to 22 overs per side, McGrath taking 3/16 off 5
v England won by 7 wickets with 16 balls to spare 248/3 47.2 5.23 248 2 Pietersen’s 104 set up a good score, but Ponting’s 86 helped see off the target.
v Ireland won by 9 wickets with 226 balls to spare 92/1 12.2 7.45 92 2 Ireland dismissed for just 91 with McGrath taking 3/17 off 7
v Sri Lanka won by 7 wickets with 44 balls to spare 232/3 42.4 5.43 227 2 Bracken’s 4/19 off 9.4 overs helped restrict Sri Lanka, 60s by Ponting and Symonds finished the job.
v New Zealand won by 215 runs 348/6 50 6.96 1 A huge win set up by Hayden’s 103 and Watson’s unbeated 65 off 32 balls, before 4 wickets by Hogg and 3 by Tait.
v South Africa won by 7 wickets with 111 balls to spare 153/3 31.3 4.85 150 2 South Africa’s top order reduced to 5/27 with Tait (4/39) and McGrath (3/18 off 8 ) doing the damage.
v Sri Lanka
(2007 final)
won by 53 runs 281/4 38 7.39 1 Set up between overs 11 to 23 in which Australia scored 123 runs, Gilchrist ending up with 149 off 104 balls.
v Zimbabwe won by 91 runs 262/6 50 4.85 1 Watson top scored with 79 and Johnson took 4/19.
v New Zealand won by 7 wickets with 96 balls to spare 207/3 34 6.09 206 2 Johnson and Tait with 7 wickets between them, the Australian openers then added 133 in the first 18 overs.
v Sri Lanka no result A storm reached the ground when Sri Lanka were 146/3 in 32.5 overs.
v Kenya
won by 60 runs 324/6 50 6.48 1 Half centuries by Clarke (93 off 80), Haddin (65 off 79) and Hussey (54 off 43), Collins Obuya left on 98 not out for the Kenyans.
v Canada won by 7 wickets with 91 balls to spare 212/3 34.5 4.85 211 2 The teams played for the first time since 1979. Lee took 4/46 and 183 added for Australia’s first wicket.

Only a few games were even close: a couple in each of the 1999 and 2003 World Cups. Can’t wait to see how this year’s World Cup plays out!

Here are highlights from possibly my favourite World Cup game, the 1999 semi-final between Australia and South Africa.

Finally, one of my favourite World Cup innings, by Adam Gilchrist in the 2007 final against South Africa.

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