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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How to Derive the Characteristic Polynomial of a Companion Matrix?

I am working on a problem involving the characteristic polynomial of a companion matrix and need some help understanding the derivation. Here is the matrix in question: $ C(p) = \begin{pmatrix} 0 &...
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How to solve a differential equation of the 3rd order without factorisation

I'm just starting with differential equations and I'm having trouble solving the following one: $\dddot{y} + m\ddot{y} - 2m^3y = 0$ I've determined the characteristic polynomial to be $\lambda^3 + m\...
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Is the Jordan normal form uniquely determined by the characteristic and minimal polynomial if they are the same?

I'm studying for an exam and I can't get anywhere with a problem. I've seen similar questions on here but not the same. The problem provides the characteristic polynomial $XA(x) = (x-3)^2(x-1)$ and ...
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Faster way to find the eigenvalues of a 4x4 real matrix?

I want to calculate the eigenvalues and eigenspaces of this matrix for self-study: $\frac{1}{31}\left( \begin{array}{rrr} 43 & 9 & -23 & -61\\ 16 & -19 & -10 & 22 \\ 130 & ...
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Finding characteristic polynomial of a square matrix and how to proving that matrix is diagonalizable [closed]

Let $A$ be a square matrix of order $n$ such that $|A + I| = |A − 3I| = 0$ and also $\operatorname{rank}(A)= 2$. I need to find characteristic polynomial of $A$ and have prove that $A$ is ...
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Simplifying eigenvalue calculation where only one non-zero element is shared by a row and column.

I'm trying to determine the eigenvalues for the following matrix: $$ \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -...
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Determine the entries $x$ and $y$ in a matrix so that its only eigenvalue is $1$.

I am doing some self-study in preparation for an exam, and in this problem I am given the following matrix in $R^{3×3}$: $\begin{pmatrix} 1&0&1\\ 0&1&-1\\ 0&x&y\\ \end{pmatrix}$...
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Does similarity of matrices preserve sum of principal minors?

I am new to linear algebra; someone on Quora posted this "shortcut" method to find the characteristic equation of a $3\times 3$ matrix. Though they demonstrated it through an example, here's ...
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A $4\times 4$ matrix counterexample. [duplicate]

A question in Dummit & Foote is asking to prove that two $3\times 3$ matrices are similar iff they have the same characteristic and the same minimal polynomial. I was able to prove that. But then ...
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Confusion about the primary decomposition theorem in linear algebra

I am currently studying the Jordan canonical form which uses the primary decomposition. I have seen the generalised eigenspace decomposition and I know that the algebraic multiplicity which appears in ...
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What are the eigenvalues of a particular type of block partitioned matrix

Let $C=\begin{bmatrix} A & \pm J \\ \pm J^T & B\end{bmatrix}$ be a square block partitioned matrix of order $m+n$ where $A$ and $B$ are square symmetric matrices of orders $m$ and $n$ ...
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Spectrum of a strongly regular graph with a vertex deleted

I want to know if it is possible to calculate the characteristic polynomial of a strongly regular graph denoted $SRG(n,k,\lambda,\mu)$ when one or two of its vertices are deleted. I have found some ...
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Classifying Matrices

Determine up to similarity all $3 \times 3$ complex matrices $A$ such that $A^4 + 2A^3 + A^2 = 0$ and $A^2 + A \neq 0$. Give the characteristic and minimal polynomial of each matrix. I'm not quite ...
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$A,B\in M_n(\mathbb{R})$ ($n\geq 2$) are such that $A^2=-I_n$ and $AB=BA$. Can we infer that $\det{B}\geq0$? [duplicate]

Suppose that $n\geq 2$ and that $A,B\in M_n(\mathbb{R})$ are such that $A^2=-I_n$ and $AB=BA$. Can we infer that $\det{B}\geq0$? My attempt: Since $A^2=-I_n$, we have $A^{-1}=-A$ and $\det{A}=\pm1$. ...
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Characteristic polynomial of convex combination

Given the following lemma on rank-$1$ matrices which I think I understand How could I deduce the following on the characteristic polynomial of $A_s$? I don't get it just using multilinearity of ...
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A doubt on existence of a linear tranformation with some property

Let $U$ and $V$ be the subspaces of $\mathbb{R}^3$ defined by $U=\{(x,y,z)^T:2x+3y+4z=0 \}$ and $V=\{(x,y,z)^T:x+2y+5z=0 \}.$ Then consider the following statement: There exists a linear ...
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$Tr(A^k) = Tr(B^k)$

Let $A,B \in \mathcal{M}_n(\mathbf{C})$. Let us consider the following assertions : (i) $\forall k \in [[ 1,n ]]$, $\mathrm{Tr}(A^k) = \mathrm{Tr}(B^k)$ (ii) $\forall k \in \mathbf{N}$, $\mathrm{Tr}(A^...
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$\det(A-\lambda I)$ or $\det(\lambda I - A)$? [duplicate]

I have noticed that in many book, given a square matrix of order $n\times n$ called $A$, the characteristic polynomial is given by $\chi_A(\lambda)=\det(A-\lambda I)$. Some university lecturers ...
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Finding all possible Jordan forms from the Characteristic polynomial

Let A be a 7 x 7 matrix with characteristic polynomial $(t − 2)^4(3 − t)^3$. It is known that in the Jordan form of A, the largest blocks for both the eigenvalues are of order 2. Show that there are ...
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Irreducible representation of the cyclic group over $\mathbb{Q}$

I'm trying to prove the following: Let $G$ be a cyclic group of order $n$. For each divisor $d$ of $n$, denote by $G_d$ the subgroup of $G$ of index $d$. Show that $G$ has an irreducible ...
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Cycles of a graph may determine the characteristic polynomial of the adjacency matrix.

I am seeking proof of the following point. Any reference or direct proof would be appreciated. Let $H$ be a directed graph, and denote by $\mathcal{H}_i$ the set of all subgraphs of $H$ with exactly $...
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Characteristic polynomial for a $4\times 4$ matrix

In a field $\mathbb{F}_4 = \{0, 1, a, a+1\}$ where $1 + 1 = 0$ and $a^2 = a + 1$, I am to find the characteristic polynomial of $A = \begin{pmatrix} a+1 & 0 & 0 & a \\ 0 & 1 & 0 &...
rawestan's user avatar
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Counting (0,1)-matrices with a given characteristic polynomial

Let $f_n\colon \{0,1\}^{n-1} \to \mathbb{N}$ be a function where $f_n(c_{n-1}, c_{n-2}, \dots, c_1)$ is the number of $n\times n$ matrices with coefficients in $\{0,1\}$ whose characteristic ...
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Relation Between Elementary Operations, Similarity, and Characteristic Polynomial

I have a linear algebra question that is really rather simple. I must have a mistaken assumption or understanding, I just don't know where. Now I have been told that matrix similarity is a ...
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Characteristic Polynomial not needing an Ordered Basis of Eigenvectors

Linear Algebra 4$_{th}$ ed. defines the characteristic polynomial for a linear operator as follows: Let T be a linear operator on a n-dimensional vector space V with order basis, $\beta$. We define ...
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Hyperbolic toral automorphism, periodic points and irreducibility of its characteristic polynomial.

Let $A = SL(n,\mathbb{Z})$ be a matrix and $\mathbb{T}^{n} = \mathbb{R}^{n}/\mathbb{Z}^{n}$ be the $n$-dimensional torus. If we assume that none of the eigenvalues of $A$ are roots of unity, then $A$ ...
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Minimal Polynomial Degree Inequality for Linear Operator and Its Square

I found a problem I got stuck with: Let $f \in \operatorname{End}(V), V$ be a finite-dimensional $F$-vector space, $F$ algebraically closed. (a) Let $ a \in F $ and $k \in \mathbb{N}$. Show: If $(f-a)^...
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Explicit unique solution formula for linear constant coefficient recurrence relation depending on a parameter when eigenvalues change multiplicity

Consider the three-term recurrence relation on $y^{n}=y^{n}(\theta) \in \mathbb{C}$ depending on the parameter $\theta$: $$ y^{n+1}(\theta) = \alpha_{2}(\theta) y^{n}(\theta) + \alpha_{1}(\theta) y^{n-...
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Does this proof of "$AB$ and $BA$ have the same characteristic polynomial" assume $k$ algebraically closed?

I am looking at Theorem 1.3.22 of this book. I was wondering whether it makes the assumption that the matrices are in an algebraically closed field $k$, and if so, where does it use such assumption.
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How to find the characteristic polynomial of the given matrix?

I am stuck at finding the characteristic polynomial of the following matrix. $$A=\begin{bmatrix} \ell a^2 & a & a & a&\dotso & \dotso& a &a \\ a & t &0&...
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Reading off module properties from the companion matrix

Let $P\in \mathbb{F}[x]$ be a monic polynomial of degree $n$ over a field $\mathbb{F}$, and $M_P$ its companion matrix. The matrix $M_P$ gives a module of $\mathbb{F}[x]$ on $\mathbb{F}^n$, by letting ...
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Find the characteristic equation of $A$

It is well-known if $\lambda$ is an eigen value of a square matrix $A$ of order $n\times n$ then $\lambda^k$ will be eigen value of $A^k$ for every positive integer $k$. Also, if $f(x)$ be a ...
User1111's user avatar
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1 answer
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Characteristic polynomial for the composition of a linear transformation with itself

Suppose I know the characteristic polynomial of the linear transformation $T$, can the characteristic polynomial of $L= T\circ T$ be obtained? At the moment I don't know any theorem or result that can ...
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Calculating the characteristic polynomial of a block tridiagonal Toeplitz Symmetric matrix

I am trying to calculate the characteristic polynomial of a block tridiagonal matrix and I need some help. This matrix is a representation of a tight-binding Hamiltonian of a finite grid of graphene, ...
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1 answer
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Given Ker(T) and Ker(T-2I), is T diagonalizable?

I'm stuck on this problem that I found on my book of linear algebra. Be $T:\mathbb{R}ˆ3 \rightarrow \mathbb{R}ˆ3$ a linear map such that $$Ker(T-2I) = \{(x,y,z) | x+y=0\} $$ and $$Ker(T) = \{ (x,y,z)|...
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Characteristic polynomial and determinant of block matrices

If the square matrix $A$ is invertible, $$\det\begin{bmatrix}A&B \\ C&D\end{bmatrix} = \det(A) \det\left(D-CA^{-1}B\right)$$ Using this formula, can we come up with the characteristic ...
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Why are the kernel of the irreducible factors of the minimal and charateristic polynomial equal?

If $\varphi$ is an Endomorphism over $V$ with $\mu_\varphi=\prod^n_{i=1}p_i^{m_i}$ for some $n,m_i$ and irreducible $p_i$ and $\chi_\varphi=\prod^n_{i=1} p_i^{c_i}$ with $c_i\geq m_i$ then $\...
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Classification of all similar $3\times 3$ matrices over $\mathbb{R}$

I want to ask a question regarding classes of similar $3 \times 3$ matrices. We were told that two matrices are similar if and only if they have the same Jordan normal form. That led me to trying to ...
watertrainer's user avatar
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1 answer
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Error in the book Graph Theory: Calculate this determinant/characteristic polynomial

In the book "Graph Theory" - by Bondy and Murty exercise 1.1.22 b) ii) wants me to prove that $$det(J-(1+\lambda)I_n) = (1+\lambda-n)(1+\lambda)^{n-1}$$ where $J$ is the $n$ x $n$ matrix ...
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Let $A \in M_n(\mathbb R)$ such that $A^2+A+5I_n = 0$. Find the characteristic polynomial of $A$ [duplicate]

Let $A \in M_n(\mathbb R)$ such that $$A^2+A+5I_n = 0$$ Find the characteristic polynomial of $A$. I tried two different approaches and got stuck on both. I am wondering if I was even headed in the ...
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Proof of Cayley-Hamilton theorem over any field $\Bbb K$

I'm currently studying the Cayley-Hamilton theorem for an exam, and I do not quite get the proof presented in the lecture. It was structured as follows: first we'll prove it over $\mathbb{C}$ using ...
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If two matrices have the same characteristic polynomial then do they have the same determinant?

There is a similar question here, but it's asking that if two matrices have the same characteristic polynomial then are they similar. If the answer were positive then the answer to my question will ...
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Let $a$ be a sequence where $a_0 = 21$, $a_1 = 35$, and $a_{n+2} = 4a_{n+1} − 4a_n + n^2$ for $n ≥ 2$. Compute $a_{2006} \pmod {100}$

Problem Let $a_0, a_1, a_2, ...$ be a sequence of real numbers defined by $a_0 = 21, a_1 = 35$, and $a_{n+2} = 4a_{n+1}-4a_n+n^2$ for $n ≥ 2$. Compute the remainder obtained when $a_{2006}$ is ...
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Dimension of image(T) and characteristic polynomial

Let $V$ be a finite dimensional vector space and let $T: V \to V$ be a diagonalizable linear map with characteristic polynomial $$f ( x ) = x^2(x-3)^2(x-9)(x+2)$$ Find the dimension of the image of T. ...
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An $n\times n$ matrix, $n\ge 2$ with characteristic polynomial $x^{n-2}(x^2-1)$ [duplicate]

$A$ is an $n\times n$ matrix, $n\ge 2$ with characteristic polynomial $x^{n-2}(x^2-1)$. Then, which of the following is true? $A^n=A^{n-2}$ rank of $A$ is $2$ rank of $A$ is atleast $2$ there are ...
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minimal polynomial of finite field

$\frac{d}{d Y}: K[Y] \rightarrow K[Y], \quad \sum_{i \geq 0} a_{i} Y^{i} \mapsto \sum_{i \geq 0}(i+1) a_{i+1} Y^{i}$ $P_n =\[f \in K[Y] \mid \operatorname{deg}(f) \leq n]$ (this is a set) $f=\left. \...
Marius Lutter's user avatar
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1 answer
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Given a polynomial $P(x)$, is it always possible to construct a matrix with characteristic polynomial $P$? [duplicate]

Suppose you are given an arbitrary polynomial $P$ of degree $n$, with real coefficients. My question is: is it always possible to construct a square matrix of order $n$ s.t. $P$ is the characteristic ...
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Prove that the following statements are equivalent.

Let A be a nxn Matrix whose characteristic polynomial XA can be decomposed into a product of linear factors. A^n = 0 A is nilpotent 0 is the only Eigenvalue of A. I did the steps 1) => 2) but I'm ...
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1 answer
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Question related to eigenvalues and characteristic polynomial [closed]

I have this question for homework that I've been trying to solve and I couldnt understand.. happy if you could help $A$ is a square matrix with the characteristic polynomial $(t-3)^5$. It is also ...
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Find a minimal polynomial of some matrix given a minimal polynomial of another matrix.

I would like an approach on how to approach questions where I'm given a minimal polynomial $m_B(\lambda)$ (minimal polynomial of matrix B) and I'm asked to find the minimal polynomial of the matrix $2(...
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