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๐Ÿš Subject Study/๐ŸŸง Linear Algebra

์„ ํ˜•๋Œ€์ˆ˜ํ•™ 5.2

by UKHYUN22 2021. 10. 25.
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Matrix์˜ eigenvalue๋ฅผ ์ฐพ์œผ๋ผ๋Š” ๋ฌธ์ œ

1. 5.1์ ˆ ์ •๋ฆฌ๋ฅผ ์‚ฌ์šฉํ•ด์„œ Triangle์„ ๋งŒ๋“ ๋‹ค.

2. Diagonal์— ์žˆ๋Š” ์ˆ˜๊ฐ€ eigenvalue๊ฐ€ ๋˜์ง€ ์•Š์„ ๊นŒ ์ƒ๊ฐ์„ ํ•ด๋ณธ๋‹ค.

 

ํ•ด๋‹น Matrix๋Š” Invertibleํ•˜๋ฉด ์•ˆ๋œ๋‹ค. ์ฆ‰ Determinant๊ฐ€ 0์ด ๋˜์–ด์•ผ ์—ญํ–‰๋ ฌ์ด ์กด์žฌํ•˜์ง€ ์•Š๊ฒŒ ๋˜๋ฏ€๋กœ

Det์ด 0 ์ด ๋˜๋„๋ก ํ•˜๋Š” ์‹œ๊ทธ๋งˆ(T)๋ฅผ ๊ตฌํ•œ๋‹ค.

 

 

Determinant์˜ ์„ฑ์งˆ์„ ๊ธฐ์–ตํ•˜๋Š” ๊ฒƒ์ด ์ค‘์š”ํ•˜๋‹ค.

a๋ฒˆ์€ Det์ด 0์ด ์•„๋‹Œ ๊ฒฝ์šฐ์—๋งŒ ์—ญํ–‰๋ ฌ์ด ์กด์žฌํ•œ๋‹ค๋Š” ์„ฑ์งˆ.

b๋ฒˆ์€ ํ–‰๋ ฌ์˜ ๊ณฑ์˜ Det์€ ๊ฐ๊ฐ์˜ ํ–‰๋ ฌ Det์˜ ๊ณฑ๊ณผ ๋™์ผํ•˜๋‹ค๋Š” ์„ฑ์งˆ.

c๋ฒˆ์€.... A์˜ Transformation์„ ํ•œ ๊ฒƒ์˜ Det๊ณผ A์˜ Det์ด ๋™์ผํ•˜๋‹ค๋Š” ์„ฑ์งˆ.

d๋ฒˆ์€ ๋งŒ์•ฝ A๊ฐ€ ์‚ผ๊ฐ ํ–‰๋ ฌ์ธ ๊ฒฝ์šฐ Det์˜ ๊ฒฝ์šฐ ๊ฐ๊ฐ์˜ diagnol์— ์žˆ๋Š” ์ˆ˜์˜ ๊ณฑ์ด๋ผ๋Š” ์„ฑ์งˆ.

e๋ฒˆ์€ Row์˜ ๊ตํ™˜์€ Det์˜ ๋ถ€ํ˜ธ์— ์˜ํ–ฅ์„ ๋ฏธ์น˜๊ณ  Scaling์€ Det์— ์ƒ์ˆ˜ ๋ฐฐ ๋งŒํผ ์˜ํ–ฅ์„ ๋ฏธ์นœ๋‹ค๋Š” ์„ฑ์งˆ.

 

 

Characteristic Equation ์ฆ‰, ํ•ด๋‹น Det ๊ฐ€ 0์ด ๋œ๋‹ค๋Š” ์„ฑ์งˆ์„ ๋งŒ์กฑํ•ด์•ผ eigienvalue๋ฅผ ๊ฐ–๋Š”๋‹ค๋Š” ์˜๋ฏธ.

 

Characterisic Equation์„ ์ฐพ์•„๋ผ.

์šฐ์„  det(A-TI) = 0์ด ๋˜๋Š” ๊ฒƒ์„ ์ฐพ์•„์•ผ ํ•˜๋ฏ€๋กœ ์‹์„ ๋ณ€ํ˜•ํ•ด์„œ ์™„์„ฑ์‹œํ‚จ๋‹ค.

det( A -TI )๋ฅผ Characteristic polynomial ์ด๋ผ๊ณ  ๋ถ€๋ฅด๊ธฐ๋„ ํ•œ๋‹ค.

 

Characteristic Polynomial ์˜ ๋œป ์•Œ๊ธฐ.

๊ทธ๋ฆฌ๊ณ  ์ด๋•Œ Eigenvalue์™€ multiplicities๋ฅผ ๊ตฌํ•˜๊ธฐ 

๊ฐ๊ฐ์˜ Factor ์ผ์ฐจ ์‹์˜ ๊ณฑ์œผ๋กœ ํ‘œํ˜„๋˜๋„๋ก ๊ณ ์น˜๊ณ  ๋‹ต์„ ์ž‘์„ฑํ•œ๋‹ค.

 

์„ ํ˜•๋Œ€์ˆ˜ํ•™์—์„œ Similarity ๋ผ๋Š” ๊ฒƒ์€ ํ˜•ํƒœ์  ์œ ์‚ฌํ•จ์ด ์•„๋‹ˆ๋ผ ๋™์ผํ•œ Characteristic polynomial์„ ๊ฐ€์ง์— ๋”ฐ๋ผ

 ๋™์ผํ•œ eigenvalue๋ฅผ ๊ฐ€์ง€๋Š” ๊ฒƒ์„ ๋งํ•œ๋‹ค.

( ๋™์ผํ•œ Eigenvalue๋งŒ์„ ๊ฐ€์ง„๋‹ค๊ณ  Similarity ํ•œ ๊ฒƒ์€ ์•„๋‹ˆ๋‹ค.)