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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Do conjugated cyclic matrices have the same cyclic vectors?
Let $F$ be a field and $A\in GL(n,F)$ a cyclic matrix, hence there exists a $v \in F^n =: V$ such that $v, Av, ..., A^{n-1}v$ span $V$. Now let $B \in GL(n,F)$ be a matrix which is similar to $A$, so ...
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Characteristic polynomials identity as corollary of a general identity
Well, given two matrices $A,B$ of size $k\times n$ and $n\times k$ the characteristic polynomials of $AB$ and $BA$ are related, i.e., $\chi_{BA}=\lambda^{n-k}\chi_{AB}$. There are a lot of nice proofs ...
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Functor from Linear Operators to Characteristic Polynomials
I was wondering if there was any cool category theory relating to characteristic polynomials.
We can map linear transformations to their characteristic polynomials, but we of course can't map vector ...
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Distribution of coefficients of characteristic polynomial of a random matrix
Let $A \in \mathbb{R}^{n \times n}$ be a random matrix whose coefficients $a_{ij}$ are uniformly distributed in $[-1, 1]$. Writing its characteristic polynomial as
$$ p_A(x) = \det(A-xI) = \sum_k b_k ...
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Does the equation of eigenvalues tell the equation of the matrix?
If $\lambda$'s (eigenvalues) satisfy an polynomial equation, can I derive the same equation for the matrix?
To be specific, if I have $\lambda^2 = 1$, can I get $A^2 = I$?
I know the converse is ...
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$f_A(x)$ is the characteristic polynomial of $A$. $\phi_A(x)$ is the minimal polynomial of $A$. $f_A(x)/\phi_(x)$ has some property. (Ichiro Satake.)
I am reading "Linear Algebra" by Ichiro Satake.
I want to know how to prove the fact that the author wrote in Remark below.
My attempt:
$f_A(x)E=(xE-A)B(x)=(xE-A)\psi(x)B'(x)$.
$\phi_A(x)E=(...
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Given matrices $A$, $C$ such that $ACA=0$, show that characteristic polynomial of $AB$ and $A(B+C)$ is same for any matrix B. [duplicate]
Given matrices $A$ and $C$ such that $ACA=0$, show that characteristic polynomial of $AB$ and $A(B+C)$ is same for any matrix B.This question was in a worksheet for Cayley Hamilton Theorem. The order ...
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Proving a specific solution vector of the characteristic polynomial is a cyclic vector in linear algebra
I am working on a problem in linear algebra involving the characteristic polynomial of a matrix. The problem I encountered is as follows:
Let $A$ be an $n \times n$ matrix over a field $K$, with the ...
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Coefficients of the characteristic polynomial and positive definite matrices
I revisited my old notes and saw that my former tutor once told us in Linear Algebra that if we want to check if a matrix $\bf A$ is positive definite, then we can check the coefficients of the ...
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Characteristic polynomial of an orthogonal projection
Q: What is the characteristic polynomial of an orthogonal projection onto a (two-dimensional) plane through the origin in $\mathbb R^4$?
Ans: $x^2(x-1)^2$
Can someone please explain how to do this ...
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Why is $x^2+x+1$ a factor of the minimal polynomial over $\Bbb R$ just because $x^2+x+1$ is a factor of the characteristic polynomial? [duplicate]
I was studying minimal polynomials in a Linear Algebra course. I am using the book Linear Algebra by Stephen H Friedberg, Insel, and Spence for this purpose.
I was doing a problem but while reading a ...
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What do I do when the first column of my characteristic equation for finding eigenvectors are zero?
$$A=\begin{pmatrix} 2 &4 &1\\
6 &5 &2\\
3& 1& 0\end{pmatrix}$$
I got the eigenvalues to be $-2,3,-1$ and I've gotten the eigenvectors of the first $2$ eigenvalues. I'...
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Prove that the minimal and the characteristic polynomial of a linear operator are the same
Suppose $V$ is an $n$ dimensional vector space over a field $F,$ and $B=\{v_1,v_2,...,v_n\}$ be an ordered basis. Let $T:V\to V$ be the linear operator such that $T(v_1)=v_2, T(v_2)=v_3,...,T(v_{n-1})=...
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Solving $c_{n} - 2c_{n-1} - 5c_{n-2} - c_{n-3}=0$
Solve the following homogeneous recurrence relation \begin{equation} c_{n} - 2c_{n-1} - 5c_{n-2} - c_{n-3} = 0 \quad\quad n \geq 3 \end{equation} with initial conditions $c_{0} = 0, c_{1} = 1$ and $c_{...
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How to find a matrix, characteristic and minimal polynomial of a linear operator?
Given linear operator: $L : \mathbb{R}^3[X] → \mathbb{R}^3[X]$
$$L(g(X))=g(0)(-1-2X+4X^2)+g^`(1)(-1+2X^2)+g^{``}(0)(-X+\frac{1}{2}X^2)$$
Find the matrix of operator L, characteristic and minimal ...
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Simultaneous triangularisability
I have a question concerning simultaneous triangularisability (i. e. if a family of matrices $M_{\lambda}$ with $\lambda \in K$ commute and if there is one matrix $S \in GL(n, K)$ for all $M_{\lambda}$...
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Linear algebra question - what is the definition of a “distinct” eigenvalue? [closed]
I am studying university-level introductory linear algebra. This is a basic question, but I have been looking around on this and have run into nothing but confusion.
For example, the two replies to ...
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Let $X$ and $Y$ be two matrices of order $3\times 2$ and $2\times 3$. Then Least value of $|YX|$ is
Let $X$ and $Y$ be two matrices of order $3\times 2$ and $2\times 3$ respectively such that
$XY=\begin{bmatrix}
2 & -2 & 0\\
-2 & 2 & 0\\ 0 & 0 &2\\
\end{bmatrix}$
Then ...
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Let $X$ ,$Y$ be two $n\times n$ real matrices such that $XY=X^2+X+I$.
Let $X$ ,$Y$ be two $n\times n$ real matrices such that $XY=X^2+X+I$.Which of the following statements are necessarily true?
1.$X$ is invertible.2.$X+I$ is invertible.3.$XY=YX$.4.$Y$ is invertible.
...
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Find two different matrices that share characteristic and minimal polynomial
Theorem: If $A$ is a matrix in Jordan Normal Form:
• The eigenvalues of $A$ are the scalars on the diagonal of $A$.
• The algebraic multiplicity of an eigenvalue $\lambda$ is the number of times $\...
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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 &...
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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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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 $(f-a)^...
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Does this proof of "$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 ...
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