# 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.

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### Fibonacci closed form via vector space of infinite sequences of real numbers and geometric sequences

This question is from Linear Algebra with Applications (5th edition) by Otto Bretscher. It is in section 4.1 (Introduction to vector spaces), question 60. Below is the question: Consider the ...
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### Characteristic polynomial of a matrix — algorithm

Is there a classical algorithm to compute the coefficients of the characteristic polynomial of a real matrix, for small matrix sizes (say up to $10\times10$)? Is there a specialized version for ...
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### Given the characteristic equation, how to find the determinant of a matrix

Take a look at this question: Find $\det(A)$ given that A has $p(\lambda)$ as its characteristic polynomial. $$p(\lambda) = \lambda^3 - 2\lambda^2 + \lambda + 5$$ My first step is to notice the ...
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### Solving for the closed form of recurrence relations using characteristic polynomial

I know how to find the closed form of some recurrence relations such as those that are similar to the Fibonacci Sequence. I am not sure how to solve a recurrence relation using the characteristic ...
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### Let $A$ be an $n \times n$ real matrix with $n \geq 2$ and characteristic polynomial $x^{n-2}(x^2-1)$, then

$\newcommand{\rank}{\operatorname{rank}}$Let $A$ be an $n \times n$ real matrix with $n \geq 2$ and characteristic polynomial $x^{n-2}(x^2-1)$, then $A^n=A^{n-2}$ $\rank(A) \geq 2$ $\rank(A) = 2$ ...
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### Let $A$ be a $3\times 3$ matrix with characteristic polynomial $x^3-3x+a$, for what values of $a$ given matrix must be diagonalizable.

Let $A$ be a $3\times 3$ matrix with characteristic polynomial $x^3-3x+a$. For what values of $a$ given matrix must be diagonalizable. I am talking about diagonalizability over reals. Efforts: If a ...
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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 ...
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### Finding the associated matrix of a linear transformation to calculate the characteristic polynomial

Let $T : M_{n \times n}(\Bbb R) \to M_{n \times n}(\Bbb R)$ be the function given by $T(A)=A^t$ (the transpose of $A$). I need to find the minimal polynomial and the characteristic polynomial of $T$. ...
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### Characteristic polynomial of an inverse

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 ...
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### Sufficient condition for a polynomial to be a characteristic polynomial

Let $A\in \operatorname{Mat}_{n\times n}(F),~F$ being a field, satisfies $p(x)\in F[x]$ where $\deg p(x)=n$ and $p(x)$ is a monic polynomial. Can we say $p(x)=\chi_A(x)?$
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### Do $AB$ and $BA$ have 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?
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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_\...