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.
265
questions
-1
votes
1answer
23 views
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 $A$ ...
0
votes
0answers
21 views
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) ...
0
votes
0answers
27 views
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 ...
-1
votes
0answers
27 views
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 ...
0
votes
1answer
49 views
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-\...
2
votes
3answers
57 views
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 ...
0
votes
1answer
40 views
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 ...
1
vote
0answers
48 views
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 ...
1
vote
0answers
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 ...
0
votes
1answer
78 views
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 ...
1
vote
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 ...
1
vote
0answers
16 views
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 ...
1
vote
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 ...
1
vote
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 $a$ ...
0
votes
0answers
13 views
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 $p(A)=\sum_{i=0}^{2n}a_it^i$, ...
0
votes
1answer
46 views
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 ...
0
votes
1answer
15 views
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 $(x−2)^2(x−3)^2$.
3
votes
0answers
52 views
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 ...
2
votes
0answers
30 views
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:
$$
A=\left[\begin{array}{ccc}
2 & 1 & 2 \\
-2 & -1 & ...
0
votes
2answers
38 views
Prove that characteristic polynomial divides minimal polynomial to the n
Let $M \in Mat_{n\times n}(\mathbb{C})$ be a matrix with complex coefficients, char$_M(X)$ its characteristic polynomial and $m_M(X)$ its minimal polynomial.
How do I prove that char$_M(X)$ divides $...
1
vote
1answer
32 views
Minimal and characteristic polynomials of orthogonal transformations
I am given orthogonal linear transformation $U: \mathbb{R}^4 \to \mathbb{R}^4$, represented by
$$A=\begin{bmatrix} 1/2 & 1/2 & 1/2 & -1/2 \\ 1/2 & 1/2 & -1/2 & 1/2 \\ 1/2 & ...
0
votes
1answer
26 views
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 ...
1
vote
0answers
29 views
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 ...
0
votes
0answers
12 views
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 ...
1
vote
0answers
32 views
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 ...
2
votes
1answer
40 views
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 ...
3
votes
0answers
95 views
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). ...
0
votes
2answers
39 views
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 ...
1
vote
0answers
35 views
“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 ...
0
votes
1answer
36 views
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 ...
2
votes
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 &...
1
vote
0answers
63 views
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&...
1
vote
1answer
38 views
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 ...
-1
votes
2answers
53 views
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}
...
1
vote
1answer
40 views
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 ...
1
vote
2answers
75 views
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 ...
0
votes
1answer
29 views
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 ...
0
votes
0answers
15 views
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 ...
1
vote
1answer
42 views
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$$
Please show that $G(X)=Y$ has a solution $X$ for any $Y$ $\iff$ $A$ and $B$ have no common ...
0
votes
1answer
35 views
Find the eigen value of a matrix without using the characteristic eqution
For a 3x3 matrix, how to find the eigenvalues without using its characteristic equation, if one of the eigenvalues is given.
Suppose 2 is an eigenvalue of the matrix A, and find the others without ...
2
votes
0answers
63 views
Given a 4x4 Singular Matrix. find characteristic polynomial
I'm trying to solve this issue:
Given A, a 4x4 singular Matrix. It is known that $\rho(A+2I)=2$ and $|A-2I| =0$.
Find the characteristic polynomial of A, is A similar to a diagonal matrix?
I've ...
1
vote
1answer
28 views
Expansion of the Characteristic polynomial via exterior product
Looking at wiki, I want to
$p_A (t) = \sum_{k=0}^n t^{n-k} (-1)^k \operatorname{tr}(\Lambda^k A) $
where ${\displaystyle \operatorname {tr} (\Lambda ^{k}A)={\frac {1}{k!}}{\begin{vmatrix}\operatorname ...
2
votes
1answer
123 views
Characteristic polynomial of a A agrees with its minimal polynomial if and only if all matrices that commutes with A is a polynomial of A
Let $A \in \mathcal{M}_n (\mathbb{C})$. Over $\mathbb{C}$, show that the following two statements are equivalent:
Characteristic polynomial of a $A$ agrees with its minimal polynomial;
All matrices ...
0
votes
1answer
51 views
The characteristic polynomial to this question seems unsolvable, any help? [closed]
Let $a$ is a real number, determine for all values of an orthogonal matrix P and a diagonal matrix D such that $A=PDP^{T}$
$$
A=\begin{pmatrix}
1 & a & 1\\
a & 1 & 1\\
1 & 1 & ...
0
votes
2answers
62 views
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(\...
2
votes
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 ...
1
vote
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:
$$A\in\mathbb C^{n\times n}:a_{ij}=i\cdot j ,\forall i,j=1,..,n$$
$$\text{i.e for $n=3$, } $$
$$A=\left[\begin{matrix}1 & ...
2
votes
1answer
34 views
Identity between characteristic polynomial of matrix and its derivative (using the determinant lemma)
Let $A = \text{diag}(\lambda_1,\dots,\lambda_n)$ be a diagonal $n \times n$ matrix and $J:=\textbf{1}\textbf{1}^T$ be the all-ones square matrix. Using the determinant lemma, we can write
$$\det(xI-A-...
0
votes
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:...
1
vote
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(\...
