What are the eigenvalues ​​and expansions?

Question

What are eigenvalues, vectors, and decompositions, and as an algorithm constructor, how can I use them?

EDIT: I want to know how you used it in your program so that I understand. Thanks.

Answer 1

they are used much more than matrix algebra. examples include:

• the distribution of the asymptotic state of the hidden Markov model is given by the left eigenvector associated with the eigenvalue of unity from the state transition matrix.
• , ( , "" ), , , , .
• , k n >= k k . , , , .
• (, ) ( ) , - , , , , (.. , ).
• , , , t1, t2 > t1, , . , , .

Answer 2

. , .

, .. "" . , "", 1 .

A () . . Eigendecomposition .

, , :

ax ^ 2 + 2bxy + cy ^ 2 + 2dx + 2fy + g = 0

AX ^ 2 + BY ^ 2 + C = 0

X Y .

!

Answer 3

http://mathworld.wolfram.com/Eigenvalue.html

. , : .

cheers, jrh.

Answer 4

. , , .

, , , ..

Answer 5

, , H, |x> H ,

H|x> = h|x>

H , |x> H.

( , .)

:

. , . , ( ) .