All Questions
Tagged with characteristic-polynomial cayley-hamilton
31 questions
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0
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63
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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 ...
- 11
0
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1
answer
197
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Proof of Cayley-Hamilton theorem over any field $\Bbb K$
I'm currently studying the Cayley-Hamilton theorem for an exam, and I do not quite get the proof presented in the lecture. It was structured as follows: first we'll prove it over $\mathbb{C}$ using ...
- 305
0
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0
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68
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An $n\times n$ matrix, $n\ge 2$ with characteristic polynomial $x^{n-2}(x^2-1)$ [duplicate]
$A$ is an $n\times n$ matrix, $n\ge 2$ with characteristic polynomial $x^{n-2}(x^2-1)$. Then, which of the following is true?
$A^n=A^{n-2}$
rank of $A$ is $2$
rank of $A$ is atleast $2$
there are ...
1
vote
1
answer
120
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Show that $A+B=AB+BA$ iff $\text{Tr}(A)=\text{Tr}(B)=\text{Tr}(AB)=1$
We have $A,B$ $(2×2)$ matrices with complex entries. We know $AB≠BA$. Show that $A+B=AB+BA$ if and only if $\text{Tr}(A)=\text{Tr}(B)=\text{Tr}(AB)=1$.
I tried writing $A=X+Y$ and $B=X-Y$ so we can ...
- 1,311
1
vote
2
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224
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Show that equation $\det(A+xB)=0$ has real solutions if and only if $\det(A^{2}+B^{2})\geq(\det(A)+\det(B))^{2}$
We have $A,B$ two $2×2$ matrices with real values and we know $\det(AB-BA)=0$. Show that equation $\det(A+xB)=0$ has real solutions if and only if $$\det(A^{2}+B^{2})\geq(\det(A)+\det(B))^{2}.$$
I ...
- 1,311
7
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1
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257
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Prove $\det((AB)^{n}-(BA)^{n})$ is a perfect cube.
We have $A,B$ two $3×3$ matrices with integer numbers. We know that $(AB)^{2}+BA=(BA)^2+AB$.
a) Show that $\det((AB)^{n}-(BA)^{n})$ is divisible by $det(AB-BA)$.
b) Show that if $\det(AB-BA)=1$, then $...
- 1,311
3
votes
0
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133
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Theorem 4 (Cayley-Hamilton), Section 6.3 of Hoffman’s Linear Algebra
Let $T$ be a linear operator on a finite dimensional vector space $V$. If $f$ is the characteristic polynomial for $T$, then $f(T)=0$; in other words, the minimal polynomial divides the characteristic ...
- 4,508
0
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1
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44
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Implication from definition of characteristic polynomial
I know that the characteristic function of a linear map $T:V\to V$ is defined as $\chi_T(x):=\chi_A(x)$ where $A$ is any matrix for $T$ w.r.t. some basis of $V$. I know this is well-defined as it is ...
- 477
1
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1
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197
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If $A$ commutes with $(AB - BA)^2$, is $\det(AB - BA) = 0$?
We have $A$ and $B$ are $3 \times 3$ matrices with complex numbers. We know matrix $A$ is commuting with matrix $(AB-BA)^2$. Can you show $\det(AB-BA)=0$?
I tried using some Hamilton Cayley Theorem on ...
- 1,311
2
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0
answers
118
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Step in proof of Cayley-Hamilton theorem in Steinberg's book
I am reading "Representation Theory of Finite Groups - An Introductory Approach" by Benjamin Steinberg, and making exercise 2.9. I can unfortunately not find a solution anywhere. Most of the ...
- 41
2
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1
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74
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Is there any intuitive explanation to $A^2=\operatorname{tr}(A)A-\det(A)I_{2\times 2}$ for $A \in \mathbb{R}^{2\times 2}$?
$A^2=\operatorname{tr}(A)A-\det(A)I_{2\times 2}$ for $A \in \mathbb{R}^{2\times 2}$
This equation is easy to prove by denoting $$A = \begin{bmatrix}
a & b\\
c & d
\end{bmatrix}$$ but I am ...
- 33
0
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5
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166
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Suppose that $f: V \to V$ is a $k$-linear transformation such that $f^m = 0$ for some integer $m.$ Prove that $f^n = 0.$
Here is the question I want to tackle:
Let $k$ be a field and let $V$ be an $n$-dimensional vector space over $k.$ Suppose that $f: V \to V$ is a $k$-linear transformation such that $f^m = 0$ for some ...
user965463
2
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3
answers
255
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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*}...
- 753
1
vote
1
answer
105
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Characteristic polynomial of a perturbed matrix (on the first column) as function of the original characteristic polynomial
Summary of the problem: Writing the coefficients of the characteristic polynomial of a matrix where we perturb its first column as functions of the coefficients of the characteristic polynomial of the ...
- 11
1
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2
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127
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Prove that $P^k=P$ for any $k \in \mathbb{N}$ where $1$ is the only eigenvalue of $P$ implies $P=I$.
I'm having trouble proving this, using the fact $P^k=P$.
($P \in L(V)$ where $V$ is a finite-dimensional complex vector space.)
Here's my work (I didn't use $P^k=P$, but it still looks valid to me ...
- 1,298
1
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0
answers
279
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Establish if True or False: Any polynomial of degree $n$ with leading coefficient $(-1)^n$ is the characteristic polynomial
This is an exercise from Linear Algebra by Friedberg, Insel.
The question asks to determine if True or False: Any polynomial of degree $n$ with leading coefficient $(-1)^n$ is the characteristic ...
- 5,679
3
votes
1
answer
240
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Finding polynomial to the power of 2020
I am trying to solve a homework problem, but I am stuck at a point where I don't know what am I suppose to do next.
We're given a $3 \times 3$ matrix
$$A =
\begin{pmatrix}1& 2& 2\\
2& 1&...
-1
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1
answer
275
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Matrix and its characteristic polynomial [closed]
Let $A = \begin{bmatrix} a & b \\ c & d\end{bmatrix}$.
Prove that the characteristic polynomial of A can be written as $p(\lambda) = \lambda^2 − trace(A)\lambda + det(A)$ and show that $A$ ...
- 1
1
vote
0
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39
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Interpretation of Partial Characteristic Polynomials
Let $A$ be an $n \times n$ matrix over any base ring, and let the characteristic polynomial of $A$ be given by $x^n + \sum_{i = 1}^n f_ix^{n-i}$. For any $j \in \{1, \dots, n\}$, let $f^{(j)}(x) = x^j ...
- 2,736
5
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0
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286
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Is (a version of) the Cayley-Hamilton theorem true for $\mathbb{N}\times\mathbb{N}$ matrices?
Let $A\in \mathbb{C}^{n\times n}$. The Cayley-Hamilton theorem states that if $p(x)$ is the characteristic polynomial of $A$, i.e. $p(\lambda) = \det(\lambda I-A)$, then $A$ satisfies the ...
- 8,214
0
votes
4
answers
364
views
Exercise on expression with matrices using Cayley-Hamilton theorem
For the following matrix $$A=
\begin{bmatrix}
0 & 1 & 0\\
1 & 0 & 1\\
0 & 1 & 1
\end{bmatrix}
$$
I need to use the Cayley-Hamilton theorem to calculate $(A+I_3)^{10}(A-I_3)^2+A$...
user812250
1
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1
answer
310
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Alternative way to find the eigenvalues of a matrix.
I am trying questions of Masters of Mathematics Entrance exam of my university and I am looking for an alternative solution for this question.
Find Eigenvalue of the Matrix
$$
\begin {bmatrix}
...
user775699
1
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1
answer
182
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Can we apply here the Cayley–Hamilton theorem?
We have the matrix \begin{equation*}A:=\begin{pmatrix}3 & 1 & 0 & -1& -1 \\ 0 & 2 & 0 & 0 & 0 \\ 1 & 0 & 2 & 0 & -1 \\ 0 & 0 & 0 & 2 & 0 ...
- 14.1k
0
votes
1
answer
139
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Alll the matrices $A\in M_{7x7}\left(\mathbb{C}\right)$, with characteristic polynomial is: $\left(x-1\right)^3\left(x-2\right)^4$, ...
I need to find all the matrices $A\in M_{7x7}\left(\mathbb{C}\right)$,
all I know is the characteristic polynomial is:
$$\left(x-1\right)^3\left(x-2\right)^4$$
$$\dim\:\ker\:\left(A-2I\right)=3$$
$$\...
- 1,298
2
votes
1
answer
40
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find all the matrices (which are not similiar) which fulfill this formula
I need to find all the matrices $A\in M_{4x4}\left(\mathbb{C}\right)\:$ such that:
$$A^4-2A^2+I\:=\:0$$
which means $\left(A^2-I\right)^2=0$
So I see that there is a few groups of which can give ...
- 1,298
3
votes
1
answer
174
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Ring theoretic analogue of the fact that characteristic polynomial divides a power of the minimal polynomial
Proposition. Let $R$ be a commutative ring, $n\in\mathbb N$ a natural number and let $A\in\operatorname{Mat}_{n\times n}(R)$ be an $n\times n$-matrix with coefficients in $R$. Let $P_A\in R[t]$ be the ...
- 9,921
0
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1
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115
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What is the reason for so much big prof of Cayley-Hamilton theorem in Linear Algebra? [duplicate]
As I was studying linear algebra from various books I came to know about Cayley -Hamilton theorem. It states that:
An $n×n$ matrix satisfies it's own characteristic equation.
I see in various books ...
3
votes
0
answers
129
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About the characteristic equation of a square matrix (Cayley-Hamilton theorem)
Take, for example, $A$ of order $2 \times 2$. The characteristic equation comes to be $$A^2 - (\text{tr}(A)) A + \det (A) = 0 $$ But is this the only quadratic equation that this matrix satisfies? ...
- 1,953
1
vote
3
answers
7k
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Calculate matrix by using Cayley-Hamilton theorem
Calculate matrix $B = A^{10}-3A^9-A^2+4A$ using Cayley-Hamilton theorem on $A$.
$$A = \begin{pmatrix}
2 & 2 & 2 & 5 \\
-1 & -1 & -1 & -5 \\
-2 & -2 & -1 & 0 \\...
- 459
1
vote
4
answers
7k
views
Find the $n$-th power of a $3{\times}3$ matrix using the Cayley-Hamilton theorem.
I need to find $A^n$ of the matrix $A=\begin{pmatrix}
2&0 & 2\\
0& 2 & 1\\
0& 0 & 3
\end{pmatrix}$ using Cayley-Hamilton theorem.
I found the characteristic polynomial $...
- 1,385
29
votes
4
answers
7k
views
Interpreting the Cayley-Hamilton theorem
The statement of the Cayley-Hamilton Theorem is fairly straight-forward.
I now know how to find characteristic polynomials from a given matrix (or at least a matrix with certain properties that I am ...
- 10.5k