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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Matrix and its characteristic polynomial [closed]

Let $A = \begin{bmatrix} a & b \\ c & d\end{bmatrix}$. Prove that the characteristic polynomial of A can be written as $p(\lambda) = \lambda^2 − trace(A)\lambda + det(A)$ and show that $A$ ...
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Help identifying a bifurcation

I'm investigating a dynamical system and I have come across roots the real part of which looks like this: And imaginary part looks like this: The real part (which I'm primarily interested with) ...
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A = P BP^ −1 . (i) Show that two similar matrices have the same spectrum. Is the converse true? [duplicate]

Two matrices A and B are said to be similar if there is an invertible matrix P for which A = P BP^−1 . (i) Show that two similar matrices have the same spectrum. Is the converse true? how to prove its ...
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Complex roots of the characteristic polynomial

For a linear operator $T : V \to V$, given the characteristic polynomial for $[T]_{\beta}$ with respect to an ordered basis $\beta$, do the complex roots (if any) occur in pairs? Also if I change the ...
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Trying to find the Minimal Polynomial given the Characteristic Polynomial of a Linear Map [closed]

Let $T:V \to V$ be a linear transformation (where $V$ is a vector space over a field $K$). Suppose the Characteristic Polynomial of $T$ is $C_T(x) = {(x-\lambda_1)}^{r_1}{(x-\lambda_2)}^{r_2}\dots{(x-\...
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Prove that $A$ in invertible through characteristic polynomial: $x^{500}+x^{100}-x+4$

Let $A$ be a matrix with charecteristic polynomial $$p(x)=x^{500}+x^{100}-x+4$$ Prove that $A$ is invertible. I'm very lost with this one, because I don't know how to calculate the eigenvalues, I ...
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Linear transformations of a specific characteristic polynomial. [closed]

I want to know all linear transformations having the characteristic polynomial $(x-1)^3(x+1)^2$?how can I know their number? how can I know them exactly? Is it by Jordan blocks or what? Any help will ...
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Module characteristic polynomial.

I know that if I have a matrix $A,$ the characteristic polynomial is determinant of the matrix $(A-\lambda I)$ where, $\lambda$ is an eigenvalue and $I$ is an identity matrix, and the characteristic ...
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26 views

Interpretation of Partial Characteristic Polynomials

Let $A$ be an $n \times n$ matrix over any base ring, and let the characteristic polynomial of $A$ be given by $x^n + \sum_{i = 1}^n f_ix^{n-i}$. For any $j \in \{1, \dots, n\}$, let $f^{(j)}(x) = x^j ...
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If $A^3 = 2I$, prove that matrix $A - 2^{\frac{1}{3}}I$ is not invertible

If $A$ is a square real matrix and $A^3 = 2I$, how can I prove that matrix $A - 2^{\frac{1}{3}}I$ is not invertible? I know it can be solved using the characteristic polynomial of matrix $A$, but I ...
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1answer
27 views

Is the auxiliary equation of a differential equation related to characteristic polynomial for matrix eigenvalues?

I am taking a course on differential equations and one of the topics is solving second order differentials with the help of an auxillary equation. However one thing that's been bugging me alot is that ...
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Routh Table First Column 0: Total number of RHP poles

I was doing some research and found that when a first column 0 appears but everything else is not necessarily 0 in the row, then there exists poles with nonnegative real parts (or positive real parts ...
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1answer
38 views

Find $C$ such that $C^{-1}MC$ is in Jordan normal form

I have the following matrix $M$: \begin{pmatrix} 4&-1&1\\ 2&1&2\\ 1&-1&4 \end{pmatrix} I have to find a matrix $C$ such that $C^{-1}MC$ is in Jordan normal form. I have found ...
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2answers
48 views

Determining the minimal polynomial using the characteristic polynomial of a matrix

Consider a field $K$ and an extension $K(a)$. I have seen it proven that if we define $$ f_a = a\cdot \text{Id}_{K(a)}: K(a) \rightarrow K(a) $$ as the linear function defined by multiplication by $a$ ...
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When does the characteristic polynomial of an integer symplectic matrix only have $0,1,-1$ coefficients?

Suppose $A\in Sp(2n,\mathbb{Z})$ is a $2n\times 2n$ symplectic matrix with integer entries. Let $p(A)=\det(tId-A)$ be its characteristic polynomial. Suppose $tr(A)=1$. If $p(A)=\sum_{i=0}^{2n}a_it^i$, ...
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1answer
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Coefficient of characteristic polynomials

I have a big question about the coeffecient of characteristic polynomials. Indeed I want to prove that for $$p_f(X) = X^n + a_{n-1} X^{n-1} + \cdots + a_0$$ hence $a_n = (-1)^q tr(∧ ^qf)$. I already ...
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If we're given characteristic and minimal polynomial of a linear transformation, how can we find all its possible Jordan forms?

Determine all possible Jordan forms of a linear transformation with characteristic polynomial $(x−2)^4(x−3)^3$ and minimal polynomial $(x−2)^2(x−3)^2$.
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Leading term of $\det (A + \lambda B)$

Let $A$ and $B$ be $n \times n$ matrices. If $B$ is nonsingular, then the polynomial $p(\lambda) = \det (A - \lambda B)$ is clearly the characteristic polynomial of $B^{-1}A$ times $\det B$. What is ...
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Is there a fast method to calculate the non-singular matrix for the rational canonical form?

I want to take this question as an example: Rational canonical form(Frobenius normal form) of the matrix $A$ The matrix looks like: $$ A=\left[\begin{array}{ccc} 2 & 1 & 2 \\ -2 & -1 & ...
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Prove that characteristic polynomial divides minimal polynomial to the n

Let $M \in Mat_{n\times n}(\mathbb{C})$ be a matrix with complex coefficients, char$_M(X)$ its characteristic polynomial and $m_M(X)$ its minimal polynomial. How do I prove that char$_M(X)$ divides $...
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1answer
32 views

Minimal and characteristic polynomials of orthogonal transformations

I am given orthogonal linear transformation $U: \mathbb{R}^4 \to \mathbb{R}^4$, represented by $$A=\begin{bmatrix} 1/2 & 1/2 & 1/2 & -1/2 \\ 1/2 & 1/2 & -1/2 & 1/2 \\ 1/2 & ...
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Subspace consisting of normal operators

$V$ is an $n$-dimensional inner product space. Let $L=L(V,V)$ be a vector space of all linear operators on $V$, and let $T\in V$ be a normal operator. If the char. poly of $T$ splits, show that there ...
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Computing rational canonical form

I am given that $T:V\rightarrow V$ is a linear transformation over $Q$-vector space $V$. Its characteristic polynomial is $(x^2 - 2)^5(x^2+x+1)^3$ and minimal polynomial is $(x^2 - 2)^3(x^2+x+1)$. I ...
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Generalized Eigenspace which are T-cyclic

Suppose thay $T:V\rightarrow V$ is a linear transformation on a vector space $V$ such that its characteristic polynomial splits and $K_\lambda (T) = v\in V | (T-\lambda)^m v = 0 $ for some $m\geq0$ is ...
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Minimization property on minimal/characteristic polynomial

Let $A$ be a symmetric positive definite matrix and let $x^*$ denote the solution of $A x = b$ and $\lVert x \rVert= \sqrt{x^TAx}$. Then there is a minimization property (for the conjugate gradient ...
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1answer
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rank of $E_i$ in the sum $A=\lambda_1E_1+\dots+\lambda_kE_k$ where $\lambda_i$ are all the eigenvalues of A.

I'm trying to solve the following exersice. Let $A$ be a $\nu\times\nu$ matrix with elements over a field $F$ and let $\chi_A(x)=(-1)^\nu(\lambda_1-x)^{\sigma_1}\dots(\lambda_k-x)^{\sigma_k}$ be the ...
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Characteristic polynomials of two matrices containing the same elements

I asked this question in a slightly different form and didn't receive any comments. Two real, symmetric, positive semidefinite matrices $A$ and $B$ contain the same elements (in different orders). ...
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2answers
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Prove that the characteristic polynomial $P(x)$ from $A$ and $Q(x)$ from $A^{-1}$ are related by…

I have to prove that $A$ characteristic's polynomial $P(x)$ and $A^{-1}$ characteristic's polynomial $Q(x)$ are related by: $$ Q(\lambda) = \frac{\lambda^{n} P(\lambda ^{-1})}{P(0)} $$ I've tried by ...
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“Algebraic Multiplicity” in real vector spaces

I was having a look at a post on algebraic multiplicity (Understanding algebraic Multiplicity in two ways) which mentions that given a complex vector space and any eigenvalue $\lambda$, the dimension ...
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1answer
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I need help finding the roots of a characteristic polynomial

I am trying to find the roots of characteristic polynomials. This can be difficult to do by hand, especially when I have a characteristic polynomial of the 3rd or 4th degree. I always try to factorise ...
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1answer
81 views

Find the characteristic polynomial of a $10\times 10$ matrix

Let $$A=\begin{pmatrix} 0 & 1 & ... & & & & & & ...& 0\\ 0 &0 &1 &. .. & & & & & ...&0 \\ 0& 0 &0 &...
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characteristic polynomial of matrix with several blocks

$\mathbf{M}=\begin{bmatrix} \mathbf{y}&0&0&\cdots&0&0&\mathbf{x}\\ \mathbf{I}&0&0&\cdots&0&0&0\\ 0&\mathbf{I}&0&0&0&0&0\\ 0&...
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1answer
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Finding the characteristic polynomial of this matrix

I have a question about this post. How can we know that the geometric multiplicity of the eigenvalue $\lambda=0$ is $n-1$? I get that $0$ is an eigenvalue of $A$ because $\textrm{det}A=0$, but I can't ...
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Can there be a different Eigen vector for a particular Eigen value?

Please see the photo. Here, my answer came $k$ $\begin{bmatrix} 1\\ 1\\ -1 \end{bmatrix}$ But their answer is given : $k$ $\begin{bmatrix} ...
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1answer
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How to calculate this determinant (Finding characteristic polynomial).

I am trying to calculate this determinant in order to find a characteristic polynomial of a matrix, however I couldn't see any operation that I can do to make my life easier calculating this, so I ...
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2answers
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How can we evaluate the characteristic polynomial with a matrix as the parameter?

For any polynomial p(x) = $a_0+a_1x+· · ·+a_kx^k$ and any square matrix A, p(A) is defined as p(A) = $a_0I + a_1A + · · · + a_kA^k$ . Show that if v is any eigenvector of A and $χ_A(x)$ is the ...
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1answer
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How to prove that the characteristic polynomial of this specific matrix is not the power of a linear polynomial?

I was reading an algebra paper, and the problem that appeared to me is the following: The authors defined a group $G = A \rtimes \left<x\right>$, where $A$ is a finitely generated free abelian ...
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Finding maximally defective tridiagonal matrices

Question: given the real numbers $a_1, a_2, \ldots a_n \in \mathbb{R}$ which sum to zero $\sum_i a_i = 0$, solve for positive real numbers $b_1, b_2, \ldots b_{n-1} \in \mathbb{R}^+$ such that the ...
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1answer
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How to prove $G(X)=Y$ has a solution $X$ for any $Y$ $\iff$ $A$ and $B$ have no common eigenvalues.

Let $A,B\in \Bbb C^{n\times n}$ and $G:\Bbb C^{n\times n} \to \Bbb C^{n\times n}$ defined by $$G(X)=XA-BX$$ Please show that $G(X)=Y$ has a solution $X$ for any $Y$ $\iff$ $A$ and $B$ have no common ...
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1answer
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Find the eigen value of a matrix without using the characteristic eqution

For a 3x3 matrix, how to find the eigenvalues without using its characteristic equation, if one of the eigenvalues is given. Suppose 2 is an eigenvalue of the matrix A, and find the others without ...
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Given a 4x4 Singular Matrix. find characteristic polynomial

I'm trying to solve this issue: Given A, a 4x4 singular Matrix. It is known that $\rho(A+2I)=2$ and $|A-2I| =0$. Find the characteristic polynomial of A, is A similar to a diagonal matrix? I've ...
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1answer
28 views

Expansion of the Characteristic polynomial via exterior product

Looking at wiki, I want to $p_A (t) = \sum_{k=0}^n t^{n-k} (-1)^k \operatorname{tr}(\Lambda^k A) $ where ${\displaystyle \operatorname {tr} (\Lambda ^{k}A)={\frac {1}{k!}}{\begin{vmatrix}\operatorname ...
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1answer
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Characteristic polynomial of a A agrees with its minimal polynomial if and only if all matrices that commutes with A is a polynomial of A

Let $A \in \mathcal{M}_n (\mathbb{C})$. Over $\mathbb{C}$, show that the following two statements are equivalent: Characteristic polynomial of a $A$ agrees with its minimal polynomial; All matrices ...
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1answer
51 views

The characteristic polynomial to this question seems unsolvable, any help? [closed]

Let $a$ is a real number, determine for all values of an orthogonal matrix P and a diagonal matrix D such that $A=PDP^{T}$ $$ A=\begin{pmatrix} 1 & a & 1\\ a & 1 & 1\\ 1 & 1 & ...
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Must there be an invertible matrix $Q$ such that $Q^{-1}MQ=N$?

Let $M$ be a complex square matrix such that it has the following: Minimal polynomial: $\mu(\lambda)= (\lambda-3)^2(\lambda+2)^3(\lambda-5)$ Characteristic polynomial: $\chi(\lambda)=(\lambda-3)^4(\...
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2answers
81 views

Multiplication of matrix: ABAB

I was doing this Putnam exercise: Let $A$, $B$ be $n \times n$ matrices such that $ABAB = 0$. Can we conclude that $BABA = 0$? I realized that this is not always true, it can fail to non-invertible ...
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1answer
56 views

Calculate $X_A(x) $ and $m_A(x) $ of a matrix $A\in \mathbb{C}^{n\times n}:a_{ij}=i\cdot j$ [duplicate]

Caclulate the characteristic & the minimal polynomial of the matrix: $$A\in\mathbb C^{n\times n}:a_{ij}=i\cdot j ,\forall i,j=1,..,n$$ $$\text{i.e for $n=3$, } $$ $$A=\left[\begin{matrix}1 & ...
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1answer
34 views

Identity between characteristic polynomial of matrix and its derivative (using the determinant lemma)

Let $A = \text{diag}(\lambda_1,\dots,\lambda_n)$ be a diagonal $n \times n$ matrix and $J:=\textbf{1}\textbf{1}^T$ be the all-ones square matrix. Using the determinant lemma, we can write $$\det(xI-A-...
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1answer
20 views

Possibly incomplete problem - to determine minimal polynomials of projection and reflection maps

Is this problem incomplete? I don't know how to proceed! If it's incomplete, please let me know what details need to be added. Let $V=U\oplus W$. Let $P_W:V\to W$ be the canonical projection and $R_W:...
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2answers
48 views

Finding matrices for which $x^2+1$ is the minimal polynomial

Show that there is no $A\in M_3(\mathbb{R})$ whose minimal polynomial is $x^2+1$, but there is $B\in M_2(\mathbb{R})$ and $C\in M_3(\mathbb{C})$ whose minimal polynomials are both $x^2+1$. $M_n(\...

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