# 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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### 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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### 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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### 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: <...
-1 votes
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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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### 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 :...