Definition:Characteristic Equation 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 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 $R \sqbrk x$.

Also see

• Definition:Characteristic Polynomial of Matrix
• Definition:Characteristic Matrix

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): characteristic equation
• 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 equation
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): characteristic polynomial
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): characteristic