Definition:Characteristic Polynomial of Matrix
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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
- 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