CHARACTERISTIC EQUATION

In linear algebra, the 'characteristic equation' (or 'secular equation') of a square matrix ''A'' is the equation in one variable λ
:$det(A-lambda\; I)\; =\; 0\; ,$
where det is the determinant and ''I'' is the identity matrix. The solutions of the characteristic equation are precisely the eigenvalues of the matrix ''A''. The polynomial which results from evaluating the determinant is the characteristic polynomial of the matrix.
For example, the matrix
:
has characteristic equation
:
0 &{}= det(P - lambda I) \
&{}= detegin{bmatrix} 19-lambda & 3 \ -2 & 26-lambda end{bmatrix} \
&{}= 500-45lambda+lambda^2 \
&{}= (25-lambda)(20-lambda) .
end{align}
The eigenvalues of this matrix are therefore 20 and 25.
Some shortcuts exist for low dimension matrices. For a 2×2 matrix ''A'', the characteristic polynomial can be found from its determinant and trace, tr(''A''), to be
:$det(A)-\{operatorname\{tr\}\}(A)lambda+lambda^2.$
For a 3×3 matrix, we define ''c''₂ as the sum of the principal minors of the matrix, and find the characteristic polynomial to be
:$det(A)-c\_2lambda+\{operatorname\{tr\}\}(A)lambda^2-lambda^3.$
The Cayley–Hamilton theorem states that every square matrix satisfies its own characteristic equation.

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Discrete mathematics

Discrete mathematics

In discrete mathematics, the 'characteristic equation' is used when solving recurrence problems. One can specify a recurrence relation of the form
:$t\_\{n\}\; =\; At\_\{n-1\}\; +\; Bt\_\{n-2\}\; ,!$
where the value of ''t''_''n'' is dependent on the values of ''t''_''n''−1 and ''t''_''n''−2. When solving a recurrence relation, the goal is to eliminate this dependency and derive an equation of the form
:$t\_\{n\}\; =\; c\_\{1\}\{r\_\{1\}\}^n\; +\; c\_\{2\}\{r\_\{2\}\}^n\; ,\; ,!$
where ''c''₁ and ''c''₂ are constants and ''r''₁ and ''r''₂ are the roots of the characteristic equation
:$r^2\; -\; Ar\; -\; B\; =\; 0\; ,\; ,!$
where ''A'' and ''B'' are the constants defined in the original recurrence relation.