Characteristic polynomial of a matrix

Dependencies:

  1. Eigenvalues and Eigenvectors
  2. Rank of a homogenous system of linear equations
  3. A matrix is full-rank iff its determinant is non-0

Let $A$ be an $n$ by $n$ matrix. The polynomial $p_A(x) = |xI - A|$ is called the characteristic polynomial of $A$.

The roots of the characteristic polynomial are eigenvalues of $A$.

Proof

\begin{align} & \exists x \neq 0, Ax = \lambda x = \lambda (Ix) = (\lambda I)x \\ &\iff (\lambda I - A)x = 0 \textrm{ has a non-trivial solution} \\ &\iff \operatorname{rank}(\lambda I - A) < n \\ &\iff |\lambda I - A| = 0 \\ &\iff p_A(\lambda) = 0 \end{align}

Therefore, $\lambda$ is a root of $p_A(x)$ iff $\lambda$ is an eigenvalue of $A$.

Dependency for:

  1. Degree and monicness of a characteristic polynomial

Info:

Transitive dependencies:

  1. /linear-algebra/vector-spaces/condition-for-subspace
  2. /linear-algebra/matrices/gauss-jordan-algo
  3. /sets-and-relations/composition-of-bijections-is-a-bijection
  4. /sets-and-relations/equivalence-relation
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  30. Matrices over a field form a vector space
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  58. A matrix is full-rank iff its determinant is non-0