Degree and monicness of a characteristic polynomial

Dependencies:

  1. Characteristic polynomial of a matrix

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

Then $p_A$ is a monic polynomial of degree $n$.

Proof

The determinant of an $n$ by $n$ matrix is the sum of several terms where each term is the product of $n$ elements of the matrix or the negation of the product of $n$ elements of the matrix.

Since all diagonal elements of $xI - A$ contain $x$ and the rest of the elements are constant, the degree of each term is less than or equal to $n$.

The only term which contains all $x$ is the diagonal. The term given by the diagonal is of the form $\prod_{i=1}^n (x - a_{i, i})$, which is a monic polynomial of degree $n$. Therefore, $p_A(x)$ is a monic polynomial of degree $n$.

Dependency for:

  1. Every complex matrix has an eigenvalue

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