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Questions tagged [characteristic-polynomial]

The characteristic polynomial of a square matrix is a polynomial that is invariant under matrix similarity and has the eigenvalues as roots.

Let $A$ be a square $n\times n$ matrix. Its characteristic polynomial $p(\lambda)$ is $\det(\lambda I - A)$. Some authors prefer to define it as $\det(A-\lambda I)$ instead; by multiplicity of the determinant, these definitions agree for $n$ even and agree up to a sign for $n$ odd.

The characteristic polynomial contains a lot of information about $A$. Some properties include:

  • Its constant term is equal to $\det(A)$. In particular, if $p(0)\neq 0$, $A$ is invertible.
  • The coefficient of $\lambda^{n-1}$ is the trace of $A$
  • If $p$ has $n$ distinct roots, $A$ is diagonalizable (this condition is sufficient but not necessary)

A powerful result known as the Cayley-Hamilton theorem states that every matrix satisfies its characteristic polynomial; that is, $p(A)=0$. This is deeper than it appears at first: note that this does not follow by putting $\lambda=A$, as that is an abuse of notation.

A related concept is the minimial polynomial of $A$. Put simply, the characteristic polynomial allows for repeated roots and the minimal polynomial does not.

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Determinant of a rank $1$ update of a scalar matrix, or characteristic polynomial of a rank $1$ matrix

This question aims to create an "abstract duplicate" of numerous questions that ask about determinants of specific matrices (I may have missed a few): Characteristic polynomial of a matrix of $1$'s Eigenvalues of the rank one matrix…
Marc van Leeuwen
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Do matrices $ AB $ and $ BA $ have the same minimal and characteristic polynomials?

Let $ A, B $ be two square matrices of order $n$. Do $ AB $ and $ BA $ have same minimal and characteristic polynomials? I have a proof only if $ A$ or $ B $ is invertible. Is it true for all cases?
Andy
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When are minimal and characteristic polynomials the same?

Assume that we are working over a complex space $W$ of dimension $n$. When would an operator on this space have the same characteristic and minimal polynomial? I think the easy case is when the operator has $n$ distinct eigenvalues, but what…
rigot
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Which polynomials are characteristic polynomials of a symmetric matrix?

Let $f(x)$ be a polynomial of degree $n$ with coefficients in $\mathbb{Q}$. There are well-known ways to construct a $n \times n$ matrix $A$ with entries in $\mathbb{Q}$ whose characteristic polynomial is $f$. My question is: when is it possible…
Paul Siegel
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Minimal polynomials and characteristic polynomials

I am trying to understand the similarities and differences between the minimal polynomial and characteristic polynomial of Matrices. When are the minimal polynomial and characteristic polynomial the same When are they different What conditions…
CodeKingPlusPlus
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Elegant proofs that similar matrices have the same characteristic polynomial?

It's a simple exercise to show that two similar matrices has the same eigenvalues and eigenvectors (my favorite way is noting that they represent the same linear transformation in different bases). However, to show that two matrices has the same…
Gadi A
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Interpreting the Cayley-Hamilton theorem

The statement of the Cayley-Hamilton Theorem is fairly straight-forward. I now know how to find characteristic polynomials from a given matrix (or at least a matrix with certain properties that I am unaware of!). I know that the eigenvalues of the…
The Chaz 2.0
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Coefficients of characteristic polynomial of a matrix

For a given $n \times n$-matrix $A$, and $J\subseteq\{1,...,n\}$ let us denote by $A[J]$ its principal minor formed by the columns and rows with indices from $J$. If the characteristic polynomial of $A$ is $x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$, then…
Ahia Cohen
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Do characteristic polynomials exhaust all monic polynomials?

Let $A$ be an $n\times n$ matrix, then $\mathrm{char}_A(x):=\det(xI-A)$ is a monic polynomial of degree $n$. It is called the characteristic polynomial of $A$. My question is the converse: Let $p(x)$ be a monic polynomial of degree $n$. Can we…
Syang Chen
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Is a characteristic polynomial we consider in Linear Algebra a polynomial or a polynomial function?

In Linear Algebra, we consider characteristic polynomials. Is a characteristic polynomial we consider in Linear Algebra a polynomial or a polynomial function? I think it is a polynomial function. I am reading "Introduction to Linear Algebra" (in…
tchappy ha
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What do characteristic polynomials characterize?

Let $R$ be an integral domain and $F$ a finitely generated free module over $R$. For a linear transformation $\alpha\in\operatorname{End}_R(F)$, the characteristic polynomial is \begin{equation} p_\alpha(t)=\det(t-\alpha)\in R[t]. \end{equation}…
Hui Yu
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Can we deduce the characteristic polynomial for this matrix?

Given a square $n \times n$ matrix $A$ that satisfies $$\sum\limits_{k=0}^n a_k A^k = 0$$ for some coefficients $a_0, a_1, \dots, a_n,$ can we deduce that its characteristic polynomial is $\sum\limits_{k=0}^n a_k x^k$?
LIR
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Roots of minimal and characteristic polynomials

Why is it that for matrix $A \in M_n(\mathbb{C})$ the characteristic polynomial $\chi_A(t)$ and the minimal polynomial $\mu_A(t)$ have the same roots? Since $\chi_A(t) = \mu_A(t) \cdot p(t)$ it should be easy to follow, that $\chi_A(t)$ has roots…
meinzlein
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Find the characteristic polynomial of the inverse of a matrix

Given the characteristic polynomial $\chi_A$ of an invertible matrix $A$, I'm to find $\chi_{A^{-1}}$. I can see that this is theoretically possible. $\chi_A$ uniquely determines the similarity class of $A$, which uniquely determines the similarity…
Jack M
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Determining a matrix from its characteristic polynomial

Let $A\in\mathcal{M}_{n}(K)$, where $K$ is a field. Then, we can obtain the characteristic polynomial of $A$ by simply taking $p(\lambda)=\det(A-\lambda I_n)$, which give us something like $$p(\lambda) = (-1)^n\lambda^n + (-1)^{n-1}(\text{tr }…
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