# 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) ...
0answers
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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 ...
1answer
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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 ...
1answer
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### 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 ...
1answer
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### 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 ...
2answers
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### 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 $p(A)=\sum_{i=0}^{2n}a_it^i$, ...
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 ...
1answer
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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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### 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 ...
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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### 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 ...
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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### 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(\...