Definition:Characteristic Polynomial of Matrix

Definition

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 polynomial of $\mathbf A$ is the determinant of the characteristic matrix of $\mathbf A$ over $R \sqbrk x$:

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

Also defined as

Some sources define the characteristic polynomial of $\mathbf A$ as:

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

Also see

• Definition:Characteristic Polynomial of Linear Operator

• Results about the characteristic polynomial of a matrix can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): characteristic matrix
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): characteristic polynomial