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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Eigenvalues after multiplication with permutation matrix.

Let $A$ be a diagonalizable matrix, and $P$ be permutation matrix of same size. Does $A$ and $PAP$ have the same eigenvalues (or characteristic polynomial)?
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The characteristic polynomial of this family of matrices

I'm looking at the following family of $n\times n$ matrices. The entries are 0 everywhere except above and below the diagonal. Above it takes values from $1 \to n-1$ and below from $ -n +1 \to -1$. ...
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Does the minimal polynomial and characteristic polynomial have same roots over F, for a linear operator on vectorspace V over the field F?

Actually my question is that whether the minimal polynomial and the characteristic have the same root over the field of the vectorspace or do they have the same root over any extension field of F. For ...
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minimal and characteristic polynomial of this operator [duplicate]

The following is Problem 18 from Chap8.C of Axler's Linear Algebra Done Right. Edited to add a transcription of the original problem(in the image) P18. Suppose $a_0, a_1, ...., a_{n-1} \in \mathbb{C}...
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Suppose that $f: V \to V$ is a $k$-linear transformation such that $f^m = 0$ for some integer $m.$ Prove that $f^n = 0.$

Here is the question I want to tackle: Let $k$ be a field and let $V$ be an $n$-dimensional vector space over $k.$ Suppose that $f: V \to V$ is a $k$-linear transformation such that $f^m = 0$ for some ...
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Equality of minimal and characteristic polynomial [duplicate]

I'm trying to prove that for a companion matrix $C$ of a monic polynomial $f$, the minimal and the characteristic polynomial is the same. I am attempting a proof by Induction on the degree of $f$ but ...
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What is the number of linear independent eigenvectors of a complex matrix when the characteristic and minimal polynomials are the same? [closed]

Let $A$ be an $n \times n$ matrix with entries in $\mathbb{C}$. Suppose that the characteristic polynomial of $A$ is the same as the minimal polynomial of $A$ such that $$p_{A}(t) = m_{A}(t) = (t - \...
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Better Method for factoring involving Characteristic Polynomial

I am in a linear algebra class right now, and I am reviewing for diagonalization. With my final coming up I am looking to cut down time spent on finding the eigenvalues of $A$ said matrix $A$. An ...
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Coefficients of characteristic polynomial

I am studying Halmos' Finite-dimensional vector spaces, Section 82. The following is what I understand. We can express the property of positiveness of a matrix $(\alpha_{ij})$ as $$\langle A x ,x \...
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If square matrices $A^2 + B^2 = 2AB$, then prove that $p_A(x) = p_B(x)$

Original problem statement: Let $A, B \in M_n(\mathbb{C})$ such that $A^2 + B^2 = 2AB$. Prove that for any $x \in \mathbb{C}$: $$det(A - xI_n) = det(B-xI_n)$$ Now the first observation, the equality ...
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An algorithm to compute the eigenvalues of $A + B$ recursively

Is it posible to express a function $$ Eig_n(A, B) := \{ \text{The } n\text{ eigenvalues of the } n \times n \text{ matrix } A + B \}$$ The input of $Eig$ can be either described as the eigenvalues of ...
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Finding all characteristic/minimal polynomials of a matrix

I've been given this problem which I've never encountered before, neither in class or in tutorials. It goes like so: Given $A\in M_4(\mathbb{R})$ such that for the polynomial $p(x)=(x-2)(x^2+9)$ : $p(...
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Do we always have $\chi_{K/Q,x}=\pi_x$ if $K=\mathbb Q(x)$?

If $K=\mathbb Q(x)$ for $x\in\mathbb C$. Do we always have $\chi_{K/Q,x}=\pi_x$ the minimal polynomial of $x$? I am using the following definition: $\chi_{K/Q,x}$ is the characteristic polynomial of ...
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characteristic polynomial in terms of trace and determinant for 4x4 matrices

The characteristic polynomial of a 2×2 matrix can be expressed in terms of the trace(T) and determinant(D): $$\lambda^2 - T \lambda + D = 0$$ The one for 3x3 matrix can be expressed in terms of T and ...
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"Linear algebraic" proof of Frobenius normal form

Theorem: Let $\mathcal{A}$ be a linear operator on a finite dimensional vector space $V$, there exists a basis such that $V$ can be represented by the direct sum of some $\mathcal{A}$-cyclic subspace ...
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permutation representation of the symmetric group $𝑆_𝑛$ and its trace

I have this algebra task which I have encountered problems with proving a specific identity for, Consider the permutation representation of the symmetric group $𝑆_𝑛$, which gives a group ...
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solve recurrence relation $a_{n+2}=(a_{n+1}+1)/a_n$

Let $a_1$ = 2015, $a_2$=2016. Find $a_{2017}$, given $a_{n+2}=(a_{n+1}+1)/a_n$ I have found the 2 roots of the characteristic equation $x^2-x-1=0$ : $r_1 = ((1+\sqrt{5})/2)^n$ , $r_2 = ((1-\sqrt{5})/2)...
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Jordan canonical form of $p(\alpha)$ in terms of that of $\alpha$

Let $\alpha$ be a linear transformation defined in a finite-dimensional vector space $V$ over a field $F$. If polynomials $p(x)\in F[x]$ are such that for all eigenvalues $\lambda$ of $\alpha$, $p'(\...
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Compute the rank of an orthgonal projection efficiently

Question. Someone hands you an $n\times n$ matrix and tells you that it is an orthogonal projection. Describe how to compute the rank of this matrix using at most $n$ operations of addition, ...
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Show that every symmetric matrix whose entries are calculated as 1/(n−1) has an eigenvalue of 1 [duplicate]

I want to prove the following: Every symmetric matrix whose entries are calculated as $ 1/(n -1) $ with $n$ as the size of the matrix, except for the diagonal which is 0, has a characteristic ...
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Prove $\lambda^np_{AB}(\lambda) = \lambda^mp_{BA}(\lambda)$ for non-square $A$ and $B$.

Problem. Suppose $A \in \mathcal{M}_{m,n}(\mathbb{C})$ and $B \in \mathcal{M}_{n,m}(\mathbb{C})$. Prove that the characteristic polynomials for the matrices $AB$ and $BA$ satisfy $\lambda^np_{AB}(\...
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Algebraic relation between symmetric matrix and its principal submatrix?

Let $A$ be a $n\times n$ real symmetric matrix and $B$ be $m\times m$ real symmetric matrix where $n>m$. $B$ be a principal submatrix of $A$ (i.e) (obtained by deleting both $i$-th row and $i$-th ...
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Given $\Delta_{T}(x)=m_{T}(x)=(x-\alpha)^{n}$ and $ TS = ST $ where $T,S$ are linear maps. Prove there is a polynomial $f(x)$ such that $S=f(T)$.

Problem: Let $V$ denote a vector space over $\mathbb{C}$. Let $T: V \rightarrow V$ denote a linear map such that $\Delta_{T}(x)=m_{T}(x)=(x-\alpha)^{n}$ for some $\alpha \in \mathbb{C} .$ Let $S: V \...
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Proving that all $M_{2\times 2}(\mathbb{R})$ whose c.p does not split are similar to a positve multiple of a rotation matrix.

I want to prove that every matrix in $M_{2\times 2}(\mathbb{R})$ whose characteristic polynomial does not split is similar to a positive multiple of a rotation matrix, $$ s\begin{pmatrix} \cos \theta &...
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Generalized characteristic polynomial coefficients

Given two square matrices $A,\,B$ of order $n$, of which therefore all the terms are known, let us define the following polynomial: $$ p(x) := \det(A - x\,B)\,. $$ I was wondering if in the literature ...
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Characteristic Polynomial and Group Characters

TLDR: Group characters and characteristic polynomial have a very similar function but are introduced in very different terms. Can characteristic polynomial be understood through the lens of character ...
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Relation among Chern classes

Let $E\rightarrow\mathbb{P}^2$ be a rank $r$ vector bundle over $\mathbb{P}^2$ with Chern classes $c_i = c_i(E)$. Is there any relation among $c_1$ and $c_2$ or are they in general completely ...
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Characteristic polynomial of cyclic permutation

Let $P_{\sigma}$ be a the permutation matrix of a cyclic Permutation $\sigma \in S_n$, i.e. $\sigma(n) = 1$ and $\sigma(i) = i+1$ for all $1 \leq i < n$. My solutions sheet says that the ...
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Find characteristic polynomial without representing matrix

Consider the vector space of polynomials of degree 2 or lower with real coefficient $\mathbb{P}_2(\mathbb{R})$, operator $$T(f(x)) = f''(x)+f'(x)+f(0)x^2$$ and vector $g(x) = x$. Determine the ...
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For non-constant polynomial $f(t)$, there is a linear transformation $T$ such that $\phi_T(t) = (-1)^n f(t)$

I want to prove the following theorem Let $f(t)$ be a non-constant monic polynomial of degree $n$. Then there exists a linear operator $T$ on $n-$dimensional space $V$ with the characteristic ...
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Converting a monic polynomial of degree $n$ to its companion matrix

Given a polynomial of form $$p_A = t^n + a_{n-1} t^{n-1} + \cdots + a_0$$ How can construct the companion matrix $A$ such that $\det(A-tI) = p_A$ assuming that you don't already know how $A$ should ...
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Characteristic polynomial of projection

Let $V$ be a $n$ dimensional vector space over a field $\mathbb{k}$, and let $P:V \rightarrow V$ be a linear map such that $P^2=P$. i.e., A linear map is a projection. I want to find all possible ...
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Rewriting the characteristic polynomial of a matrix

Let $A\in\mathbb{R}^{n\times n}$. Assume that $m$ is a divisor of $n$ and let $k=\frac n m$. Consider the characteristic polynomial of $A$, namely, $$ p_A(s) = \det(sI_n -A). $$ My question. Is it ...
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Minimal polynomial is lcm of minimal polynomials of invariant subspaces.

Question. I am working on Hoffman and Kunze, page 219, question 4c, they ask: Let $T$ be a linear operator on $V$. Suppose $V = W_1 \oplus \cdots \oplus W_k$ where each $W_i$ is invariant under $T$. ...
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2 answers
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Given $A=B^2$. Let $p(x)$ be the characteristic polynomial of $A$ and $q(x)$ be the characteristic polynomial of $B$. Prove $p(x^{2})=q(x) q(-x)$.

Problem: Let $n$ be an even positive integer, and let $A, B \in \operatorname{Mat}_{n}(\mathbf{R})$ such that $A=B^{2}$. Let $p(x)$ denote the characteristic polynomial of $A$, and let $q(x)$ denote ...
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Solving a PDE Via the Similarity Method Versus Another Method

The PDE is $$u_{xx} + 2u_{tt} = 0$$ I imagine that the solution will be $u(v(x,\ t))$, where $v(x,\ t) = \frac{t}{x}$. So plugging this form into the PDE and using the multivariable chain rule yields ...
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Assume that $A$ has the minimal polynomial as $(x-1)(x-2)$ and c.p. as $(x-2)^2(x-1)$

Let $$A := \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1\\ \end {pmatrix}$$ Find the number of matrices similar to $A$ whose entries are from $\mathbb{Z}/\mathbb{3Z}$. ...
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Let $ A \in M_n(\mathbb{R}) $ be a matrix which satisfies $ A^2 + A + 5I_n = 0 $. Find the characteristic polynomial $ p_A $

Problem: Let $ A \in M_n(\mathbb{R}) $ be a matrix which satisfies $ A^2 + A + 5I_n = 0 $. Find the characteristic polynomial $ p_A $ I don't really know how to find $ p_A $.We can write $ p_A = x^n + ...
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2 votes
1 answer
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Proof of a polynomial matrix equation

Consider a $2 \times 2$ matrix $$ A= \left[\begin{array}{c} 2 & 7\\ 1 & 8\end{array}\right] $$ For this matrix, or for any $2 \times 2$ matrix $A$, why does $A^2 - \mbox{tr}(A) \cdot A + \det(...
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how to solve this third degree characteristic polynomial?

(this exercise is like the previous question I've written today, but now I have a 3-by-3 matrix (with a real parameter $k$). I need to say where the matrix could be diagonalized, if it's possible. (...
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Prove $n\times n$ matrix with uniform random elements is diagonizable?

Let $X$ be a $n\times n$ matrix with $n\in\mathbb{N}^{+}$ and all off diagonal elements of $X$ following a uniform distribution $x_{ij}\sim\mathcal{U}(0,1)$ for any $i,j\in\{1,...,n\}$ with $i\neq j$. ...
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Dimension of $Hom_G(V, W)$ in terms of characteristic polynomial

Let $G$ be a (pro-)cyclic group (topologically) generated by $\phi$, and let $V,W$ be two finite-dimensional (continuous) $k$-linear semisimple representations of $G$, where $k$ is some algebraically ...
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What is the difference between $\det(zI - A)$ and $\det(A - zI)$?

When computing the characteristic polynomial of matrix $A$, is it the case that $$\det(zI - A) = \det(A - zI)$$ and that one should choose the construction that is best on a case-by-case basis? If so, ...
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How to check local stability of eigenvalues generated by a quintic characteristic polynomial?

Suppose we have a 5x5 matrix, how would we check the whether the eigenvalues are negative so that we can conclude they are locally stable? As requested in the comment by Woody3: here is the matrix: <...
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Can you find a matrix by its minimal and characteristic polynomials?

If p(x) minimal polynomial and f(x) characteristic polynomial are given, and f(x) is divisible by p(x). can I always find a matrix that has f(x) as characteristic polynomial and p(x) as minimal ...
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Matrix exponential via Cayley-Hamilton

Problem For any $t\in\mathbb{R}$ compute $\exp(A_\omega t)$, where \begin{equation*}A_\omega\triangleq\left[\begin{array}{c|c} 0_2 & I_2 \\ \hline 0_2 & \Omega \end{array}\right]\end{equation*}...
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What is the characteristic equation of $a_{n+2}+2a_{n}=0$?

So if we let $a_{n}=Cr^n$ then we have $Cr^{n+2}+2Cr^n=Cr^n(r^2+2)$. So I got that the characteristic equation $r^2+2=0$ but it should be $r^2+2r=0$ apparently. How is that?
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2 votes
2 answers
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Irreducible factors of minimal and characteristic polynomial of a endomorphism over a finite dimensional $\mathbb{F}$-vector space [duplicate]

Let $V$ be a finite dimensional $\mathbb{F}$-vector space. Suppose $L:V\to V$ is an endomorphism, whose associated matrix is $A$. Now, denote its characteristic and minimal polynomial by \begin{align*}...
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When is the proper time to use row operation to find the determinant

When trying to find the determinant for a square matrix of a normal linear map such as $T: V\rightarrow W$, it's possible to just use elementary row operation to make the square matrix become an upper ...
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Finding eigenvalues of a square matrix

I am asked to find the eigenvalues of the following matrix \begin{align} M = \begin{pmatrix} p & 0 & p \\ 0 & p & p \\ p & p & 2p \\ \end{pmatrix} \end{align} I know that :...
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