Hp 50g Graphing Calculator Instrukcja Użytkownika Pobierz pdf (Strona 372)

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Page 11-45

Eigenvalues and eigenvectors

Given a square matrix A, we can write the eigenvalue equation A⋅x = λ⋅x,

where the values of λ that satisfy the equation are known as the eigenvalues of

matrix A. For each value of λ, we can find, from the same equation, values of

x that satisfy the eigenvalue equation. These values of x are known as the

eigenvectors of matrix

A. The eigenvalues equation can be written also as (A

– λ⋅I)x = 0.

This equation will have a non-trivial solution only if the matrix (A – λ⋅I) is

singular, i.e., if det(A – λ⋅I) = 0.

The last equation generates an algebraic equation involving a polynomial of

order n for a square matrix A

n×n

. The resulting equation is known as the

characteristic polynomial

of matrix A. Solving the characteristic polynomial

produces the eigenvalues of the matrix.

The calculator provides a number of functions that provide information

regarding the eigenvalues and eigenvectors of a square matrix. Some of these

functions are located under the menu MATRICES/EIGEN activated through

„Ø.

Function PCAR

Function PCAR generates the characteristic polynomial of a square matrix using

the contents of variable VX (a CAS reserved variable, typically equal to ‘X’) as

the unknown in the polynomial. For example, enter the following matrix in ALG

mode and find the characteristic equation using PCAR:

[[1,5,-3],[2,-1,4],[3,5,2]]

1 2 ... 367 368 369 370 371 372 373 374 375 376 377 ... 886 887

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