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๋ณต์ˆ˜์ „๊ณตํ•˜๊ณ  ์žˆ๋Š” ์ˆ˜ํ•™๊ณผ์˜ ํ•™๋ถ€ ์กธ์—…์‹œํ—˜์„ ์œ„ํ•ด 2024๋…„ 10์›”๋ถ€ํ„ฐ ์„ ํ˜•๋Œ€์ˆ˜๋ฅผ ๋‹ค์‹œ ๊ณต๋ถ€ํ•˜๊ณ  ์žˆ์Šต๋‹ˆ๋‹ค. (ํ˜„์žฌ์ง„ํ–‰ํ˜•โ€ฆ ๐Ÿƒโ€โ™‚๏ธโ€โžก๏ธ) ์„ ํ˜•๋Œ€์ˆ˜์— ๋Œ€ํ•œ ์ „์ฒด ํฌ์ŠคํŠธ ๋ชฉ๋ก์€ โ€œLinear Algebraโ€œ์—์„œ ํ™•์ธํ•˜์‹ค ์ˆ˜ ์žˆ์Šต๋‹ˆ๋‹ค!

๋“ค์–ด๊ฐ€๋ฉฐPermalink

์ง€๋‚œ ํฌ์ŠคํŠธ์—์„œ ์ด์ฐจ ํ˜•์‹์— ๋Œ€ํ•ด ์‚ดํŽด๋ณด์•˜์Šต๋‹ˆ๋‹ค ใ…Žใ…Ž ์ด๋ฒˆ ํฌ์ŠคํŠธ๋„ ์ด์ฐจ ํ˜•์‹์— ๋Œ€ํ•ด ์‚ดํŽด๋ณด๋Š”๋ฐ์š”! ์ด์ฐจํ˜•์‹์—์„œ ๋ณด์ด๋Š” 2๊ฐ€์ง€ ํŠน์ˆ˜ ์ผ€์ด์Šค๋ฅผ ์‚ดํŽด๋ด…๋‹ˆ๋‹ค!

Does Complex MatrixPermalink

์ง€๊ธˆ๊นŒ์ง€๋Š” ์ด์ฐจํ˜•์‹ Q(x)๋ฅผ ์‹ค์ˆ˜ ํ–‰๋ ฌ์— ๋Œ€ํ•ด ์ •์˜ํ•˜๊ณ  ์‚ดํŽด๋ณด์•˜์Šต๋‹ˆ๋‹ค. ์ด๊ฒƒ์„ โ€œ๋ณต์†Œ ํ–‰๋ ฌโ€๋กœ ํ™•์žฅํ•˜๊ณ , ์—ฌ๊ธฐ ์œ„์—์„œ Positive Definite๋ฅผ ์ •์˜ ํ•ฉ์‹œ๋‹ค!

Positive Definite, when for AโˆˆCnร—n and for all non-zero xโˆˆRn,

Q(x)=xTAx>0

์ €๋Š” ์ฒ˜์Œ์— ๊ฐœ๋…์„ ์ดํ•ดํ•  ๋•Œ, ์ž˜๋ชป ์ดํ•ดํ•ด์„œ ์ž…๋ ฅ ๋ฒกํ„ฐ๋„ ๋ณต์†Œ ๋ฒกํ„ฐ์ธ ์ค„ ์•Œ์•˜์Šต๋‹ˆ๋‹ค ใ… ใ…  ์ž…๋ ฅ ๋ฒกํ„ฐ๋Š” ์—ฌ์ „ํžˆ ์‹ค์ˆ˜ ๋ฒกํ„ฐ ์ž…๋‹ˆ๋‹ค!

์–ด๋–ค ๋ณต์†Œ ํ–‰๋ ฌ์ด Positive Definite๋ฅผ ๋งŒ์กฑํ• ๊นŒ์š”? ์‹ค์ˆ˜ ์˜์—ญ์—์„œ Positive Definite ์˜€๋˜ ํ–‰๋ ฌ๋“ค์€ ๋ณต์†Œ ์˜์—ญ์—์„œ๋„ Positive Definite ์ผ๊นŒ์š”?

Hermitian MatrixPermalink

๋ณต์†Œ ํ–‰๋ ฌ A์ด Positive Definite๋ฅผ ๋งŒ์กฑํ•˜๊ธฐ ์œ„ํ•ด์„  ์•„๋ž˜ ์กฐ๊ฑด์ด ํ•„์š” ํ•ฉ๋‹ˆ๋‹ค.

  • ํ–‰๋ ฌ A๊ฐ€ ์ •์‚ฌ๊ฐ ํ–‰๋ ฌ
  • ํ–‰๋ ฌ A๊ฐ€ Hermitian Matrix

์ด๋•Œ, ์—๋ฅด๋ฏธํŠธ ํ–‰๋ ฌ์€ ์•„๋ž˜์™€ ๊ฐ™์ด ์ •์˜ํ•˜๋Š” ๋ณต์†Œ ํ–‰๋ ฌ ์ž…๋‹ˆ๋‹ค.

A=Aโˆ—

์ด๊ฒƒ์€ โ€œ์ผค๋ ˆ ์ „์น˜โ€๊ฐ€ ๊ฐ™์€ ํ–‰๋ ฌ์„ ๋งํ•ฉ๋‹ˆ๋‹ค! ์˜ˆ์‹œ๋ฅผ ๋“ค์–ด๋ณด๋ฉด,

1. ์‹ค์ˆ˜ ๋Œ€์นญ ํ–‰๋ ฌ์€ ์ž๋™์œผ๋กœ ์—๋ฅด๋ฏธํŠธ ํ–‰๋ ฌ์ด ๋ฉ๋‹ˆ๋‹ค!

A=[2335]

2. ์ผค๋ ˆ ์ „์น˜๊ฐ€ ๊ฐ™์•„์•ผ ํ•ฉ๋‹ˆ๋‹ค.

A=[21+i1โˆ’i5]

When inputs complex vectorPermalink

ํ–‰๋ ฌ A๊ฐ€ Positive Definite๋Š” ์ž…๋ ฅ์ด ์‹ค์ˆ˜ ๋ฒกํ„ฐ์ธ ๊ฒฝ์šฐ์— ์ •์˜ํ•˜๋Š” ๊ฒƒ ์ž…๋‹ˆ๋‹ค. ์ž…๋ ฅ ๋ฒกํ„ฐ๊ฐ€ ๋ณต์†Œ ๋ฒกํ„ฐ๋ผ๋ฉด, ์ด์ฐจ ํ˜•์‹์˜ ๊ฐ’์ด ์Œ์ˆ˜๊ฐ€ ๋  ์ˆ˜๋„ ์žˆ์Šต๋‹ˆ๋‹ค!

์˜ˆ๋ฅผ ๋“ค์–ด, ์ด๋Ÿฐ Positive Definite ํ–‰๋ ฌ A๊ฐ€ ์žˆ์„ ๋•Œ,

A=[2001]

์ด ํ–‰๋ ฌ์˜ ์ด์ฐจ ํ˜•์‹์€ ๋ชจ๋“  ์‹ค์ˆ˜ ๋ฒกํ„ฐ์— ๋Œ€ํ•ด Q(x)>0์„ ๋งŒ์กฑํ•˜์ง€๋งŒ, ๋ณต์†Œ ๋ฒกํ„ฐ์˜ ๊ฒฝ์šฐ

Q([i,0]T)=[i,0][2001][i0]=iโ‹…2โ‹…i=โˆ’2

์Œ์ˆ˜๊ฐ€ ๋‚˜์˜ต๋‹ˆ๋‹ค. ๋˜๋Š” ๋ณต์†Œ ๋ฒกํ„ฐ์—์„œ๋Š” ๋ณต์†Œ์ˆ˜๊ฐ€ ๋‚˜์˜ต๋‹ˆ๋‹ค.

๊ทธ๋ž˜์„œ Positive Definite๋Š” ํ–‰๋ ฌ A๋Š” ๋ณต์†Œ ํ–‰๋ ฌ์ผ ์ˆ˜๋„ ์žˆ์ง€๋งŒ, ์˜ค์ง ์‹ค์ˆ˜ ๋ฒกํ„ฐ๋ฅผ ์ž…๋ ฅ์œผ๋กœ ๋ฐ›๋Š” ๊ฒฝ์šฐ๋ฅผ ๊ธฐ์ค€์œผ๋กœ ์ •์˜ํ•ฉ๋‹ˆ๋‹ค!

Skew-symmetric MatrixPermalink

์ด ํ–‰๋ ฌ์€ ํ–‰๋ ฌ A๊ฐ€ ์•„๋ž˜ ์กฐ๊ฑด์„ ๋งŒ์กฑํ•ฉ๋‹ˆ๋‹ค.

AT=โˆ’A

๋ญ”๊ฐ€ ์‹ ๊ธฐํ•˜์ฃ ?? ๊ทธ๋Ÿฐ๋ฐ ํ–‰๋ ฌ์ด ์ด๊ฒƒ์„ ๋งŒ์กฑํ•˜๊ธฐ ์œ„ํ•ด์„œ๋Š” ๋ชจ๋“  ๋Œ€๊ฐ ์„ฑ๋ถ„์ด 0์ด ๋˜์–ด์•ผ ํ•ฉ๋‹ˆ๋‹ค! ์™œ๋ƒํ•˜๋ฉด, ๋Œ€๊ฐ ์„ฑ๋ถ„์€ ์ „์น˜ ๋˜์–ด๋„ ๊ทธ ์ž๋ฆฌ์— ๋‚จ์•„ ์žˆ์œผ๋‹ˆ๊นŒ์š”!

A=[03โˆ’30]

์ด๋Ÿฐ ํ–‰๋ ฌ์„ Skew-symmetric, โ€œ๋ฐ˜๋Œ€์นญ ํ–‰๋ ฌโ€์ด๋ผ๊ณ  ํ•ฉ๋‹ˆ๋‹ค.


ํ–‰๋ ฌ A๊ฐ€ ๋ฐ˜๋Œ€์นญ ํ–‰๋ ฌ์ด๋ผ๋ฉด, ์ด์ฐจํ˜•์‹ Q(x)์˜ ๊ฐ’์ด ํ•ญ์ƒ 0์ด ๋ฉ๋‹ˆ๋‹ค!

Q(x)=xTAx=xT(โˆ’AT)x=xT(โˆ’A)x(symmetric)=โˆ’Q(x)

Q(x)=โˆ’Q(x)๋ฅผ ๋งŒ์กฑํ•˜๋ ค๋ฉด, ์ด์ฐจํ˜•์‹์˜ ๊ฐ’์ด ํ•ญ์ƒ 0์ด ๋˜์–ด์•ผ ํ•ฉ๋‹ˆ๋‹ค.

์ด์ฐจ ํ˜•์‹์˜ ๊ฒฐ๊ณผ๊ฐ€ ํ•ญ์ƒ 0์ด ๋˜์–ด๋ฒ„๋ฆฌ๊ธฐ ๋•Œ๋ฌธ์—, Skew-symmetric ํ–‰๋ ฌ๋กœ ์ด์ฐจ ํ˜•์‹์„ ํ•ด์„ํ•˜์ง€๋Š” ์•Š๋Š”๋‹ค๊ณ  ํ•ฉ๋‹ˆ๋‹ค!


๋ฐ˜๋Œ€์นญ ํ–‰๋ ฌ์€ โ€œํšŒ์ „ ๋ณ€ํ™˜โ€œ์„ ์ˆ˜ํ–‰ํ•ฉ๋‹ˆ๋‹ค! ๋‚˜๋จธ์ง€ ๋‚ด์šฉ์€ To be continuedโ€ฆ