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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$ A^3 + B^2 = I_n $ and $A^5=A^2$, then $\det(A^2 + B^2 + I_n) \geq 0 $ and $\operatorname{rank}(I_n + AB^2) = \mathrm{rank}(I_n - AB^2) $

Let $A, B$ be square matrices of size $n$, $n \geq 2$, containing real entries. $\DeclareMathOperator\rank{rank}$ If the following properties take place: $ A^3 + B^2 = I_n $ and $A^5=A^2$, then $...
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1answer
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Matrix representation and characterstic polynomial of linear transformation from the 0 vector space to a finitely-dimensional space

Relevant definition: In this definition, what happens if $V = \{0\}$, the 0 vector space? The basis of the zero vector space is the empty set, so the definition would not make much sense. Do we just ...
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Characteristic polynomial in $\mathbb{C}^2$

Consider a finite-dimensional vector space $V$ over a field $\mathbb{F}$.Let $a,b\in\mathbb{F}$. State whether the following is true or not: If $\begin{pmatrix} b & a \\ 0 & b \end{pmatrix}$ ...
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Diagonalizability of a matrix

Show that $$ A :=\begin{pmatrix} 1 & 0 & 0 \\ -2 & 1 & 2 \\ -2 & 0 & 3 \end{pmatrix}$$ is diagonalizable. What I did: First, I determined the characteristic polynomial $$\...
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$ A^2 - B^2 = I_{2n+1} \implies det(AB-BA)=0 $ where A,B are complex matrices of odd size

Let $A, B$ be square matrices (with complex entries) of size $2n+1$, where $n$ is a positive integer. I need help proving the following: $$A^2 - B^2 = I_{2n+1} \implies det(AB-BA)=0 $$ I've tried ...
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Calculate matrix by using Cayley-Hamilton theorem

Calculate matrix $B = A^{10}-3A^9-A^2+4A$ using Cayley-Hamilton theorem on $A$. $$A = \begin{pmatrix} 2 & 2 & 2 & 5 \\ -1 & -1 & -1 & -5 \\ -2 & -2 & -1 & 0 \\...
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1answer
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Kernel of characteristic polynomial of linear transformation

Let $F$ be a Field and $V$ be a Vectorspace with $\dim(V)=n$. Let $f:V \to V$ be a linear transformation with the characteristic polynomial $pa(x)$. It is linear factorized. Let $pa(x)=g(x)*h(x)$ ...
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1answer
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Drawing conclusions from a characteristic polynomial

I was given a characteristic polynomial and was asked to draw some conclusions from it, but i need some help with figuring something out. My polynomial is this : $\lambda^4 -3\lambda^3 +\lambda^2 +3\...
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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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4answers
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Find the $n$-th power of a $3{\times}3$ matrix using the Cayley-Hamilton theorem.

I need to find $A^n$ of the matrix $A=\begin{pmatrix} 2&0 & 2\\ 0& 2 & 1\\ 0& 0 & 3 \end{pmatrix}$ using Cayley-Hamilton theorem. I found the characteristic polynomial $...
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$c_{n-1}$-th factor of polynomial and the trace of a matrix using induction

I've been looking around stack exchange, but I've been having some issues with this question. Let $A\in M_{n,n}(K)$, for some field $K$. We write $$c_A = (-1)^n \left(t^n + \sum_{i=0}^{n-1} c_i t^i \...
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Calculating determinant using eigenvalues (real and complex)

I was trying to find the eigenvalues for the following characteristic polynomial: $$p_M(\lambda)=(\lambda^{4}+1)(\lambda^2-4)$$ For this I solved the equation $p_M=0$ and I found the following two ...
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Finding characteristic polynomial and eigenvalues of a linear transformation

Let $T:R_3[x] \to R_3[x]$ Linear transformation such that: $$ T(ax^2 + bx + c) = (a+b+c)x^2 + (2a + 2b + 2c)x + a+b-c $$ I want to find eigenvalues for $T$. Therefore I looked at the representing ...
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Derivative of coefficients of characteristic polynomial

Let $ X(t) $ be a square matrix of dimension $ n $. The Jacobi formula expresses the derivative of the determinant of $X(t)$ in terms of the derivative of the matrix itself. Is there an analogous ...
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Monic polynomial and companion matrix

Problem Let $p(T) := T^n-\alpha_{n-1}T^{n-1}-\alpha_{n-2}T^{n-2}-\cdots-\alpha_0 \in K[T]$. Additionally we have the companion matrix of $p$ $$A:= \begin{pmatrix} 0 & 1 & 0 & 0 & \...
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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 $a_1, a_2, \dots, a_n,$ can we deduce that its characteristic polynomial is $\sum\...
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1answer
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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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How to compute the characteristic polynomial of a companion matrix to a polynomial with matrix-valued coefficients?

Consider we have a polynomial $p = z^m + b_{m-1}z^{m-1} + \dotsb + b_0$ with matrix coefficients $b_i \in M_n(\mathbb{C})$. Then we might consider the companion matrix $$T = \left[ \begin{matrix} 0_n &...
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1answer
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Question involving characteristic polynomial of a linear transformation

I was wanting some hints on a question and I have no idea how to approach this: Suppose $F$ is a field, $V$ is an $F$-vector space and $T: V \rightarrow V$ is a linear map. Suppose $p(x) \in F[x]$ ...
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Find the characteristic polynomial |$\lambda - AI $| for this $5 \times 5$ matrix

Find the characteristic polynomial |$\lambda - AI $| for the $5\times 5$ matrix $$A=\left(\begin{matrix} 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & ...
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Factorizing a polynomial.

Factorize the polynomial $$P(x)= \begin{vmatrix} a_1^2-x & a_{1}a_2 & a_1a_3 & \cdots & a_1a_n \\ a_2a_1 & a_2^2-x & a_{2}a_3 & \cdots & a_2a_n\\ a_3a_1 & ...
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How to determine the characteristic polynomial of the $4\times4$ real matrix of ones? [duplicate]

$$\left[\begin{array}{l}1&1&1&1\\1&1&1&1\\1&1&1&1\\1&1&1&1\end{array}\right]$$ I am having difficulties with calculating this. Currently I'm stuck at ...
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How to conclude that the minimal polynomial is the characteristic?

I am given the following matrix $$A=\begin{bmatrix} 0 & 0 & 4 & 1\\ 0& 0 & 1 & 4\\ 4 & 1 & 0 &0\\ 1 & 4 & 0 & 0 \end{bmatrix}$$ And I have to find the ...
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Calculate the determinant of $A-2A^{-1}$ given the characteristic and minimal polynomials of $A$

Given the characteristic polynomial and minimal polynomial of $A$ being $(x+2)^{6}(x-1)^{3}$ and $(x+2)^{2}(x-1)^{3}$, respectively. How do I determine the characteristic polynomial and minimal ...
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1answer
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Polynomials $\&$ Matrices

Assume $A$ is a matrix of order $n$. We know that the characteristic polynomial of matrix $A$ is obtained as follows $$ P(x)=\det (A-x\,I)\, . $$ Where $I$ is an identity matrix of order $n$. What ...
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Cyclic Modules, Characteristic Polynomial and Minimal Polynomial

Suppose that $\mathrm{dim}_{F}M<\infty$ for $F$ a field and $M$ an $F$ vector space. Let $T$ be a linear transformation on $M$. Show that $M$ is cyclic (as an $F[x]$ module) if and only if $m(x)$ ...
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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(\...
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Eigenvalues of a companion matrix

I've been tasked with the following: Show that the companion matrix $C(p)$ of $p(x) = x^2 + ax + b$ has characteristic polynomial $\lambda^2 + a\lambda + b$. Show that if $\lambda$ is an ...
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Characteristic polynomial of a square matrix of size $n$ with all entries equal to $1$

Let $A$ be a square matrix of size $n$ such that all the entries of $A$ are $1$. Find the characteristic polynomial of $A$.
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Eigenvalues of negative companion matrix

Here's a homework question I've been stuck on for a while. Given $$A = \begin{bmatrix} 0 & 0 & 0 & \cdots & 0 & a_0 \\ -1 & 0 & 0 & \cdots & 0 & a_1 \\ ...
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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_\...
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When is a matrix similar to the companion matrix of its characteristic polynomial?

Let $A$ be a complex matrix and $A_c$ the companion matrix of its characteristic polynomial. From what I have read, I believe the following two statements to be true: not every $A$ is similar to $...
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Minimal polynomials and characteristic polynomials [duplicate]

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 ...
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Cool/Useful Examples of Characteristic and Minimal Polynomials?

I'm teaching a Linear Algebra II undergrad course and for the section on characteristic & minimal polynomials, I really don't want to just give the students a bunch of matrices that have no ...
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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). ...
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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 ...
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Characteristic polynomial of companion matrix [duplicate]

I have a matrix in companion form, $A=\begin{pmatrix} 0 & \cdots & 0& -a_{0} \\ 1 & \cdots & 0 & -a_{1}\\ \vdots &\ddots & \vdots &\vdots \\ 0 &\cdots & ...
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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)...
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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 ...
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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 $...