Definition:Characteristic Equation

Definition

A characteristic equation is a member of the class of equations which in some way allows one to sum up a number of characteristics of a particular mathematical object.

Disambiguation

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Characteristic Equation may refer to:

Matrix

Let $R$ be a commutative ring with unity.

Let $\mathbf A$ be a square matrix over $R$ of order $n > 0$.

Let $\mathbf I_n$ be the $n \times n$ identity matrix.

Let $R \sqbrk x$ be the polynomial ring in one variable over $R$.

The characteristic equation of $\mathbf A$ is the equation defined as: determinant of the characteristic matrix of $\mathbf A$ over $R \sqbrk x$:

$\map \det {\mathbf I_n x - \mathbf A} = 0$

where $\map \det {\mathbf I_n x - \mathbf A}$ is the characteristic polynomial of the characteristic matrix of $\mathbf A$ over $R \sqbrk x$.

Differential Equation

Let:

$(1): \quad y + p y' + q y = 0$

be a constant coefficient homogeneous linear second order ODE.

The auxiliary equation of $(1)$ is the quadratic equation:

$m^2 + p m + q = 0$

Some sources refer to the auxiliary equation as the characteristic equation of $(1)$.