Questions tagged [cayley-hamilton]
For questions about the Cayley-Hamilton theorem, which states that every square matrix over a commutative ring (such as the real or complex field) satisfies its own characteristic equation.
207 questions
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Finding inverse of a $3 \times 3$ matrix with Cayley-Hamilton, and Diophantine equation
I am trying to use this formula to find the inverse of a $3 \times 3$ matrix.
$$ \mathbf A^{-1} = \frac{1}{\det(\mathbf A)} \sum_{s=0}^{n-1}\mathbf A^{s} \sum_{k_{1}, k_{2},\dots,k_{n-1}} \prod_{l=1}^{...
- 23
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Check proof that the determinant of a polynomial matrix commute with evaluation
This is used in one of the many proofs for the Cayley-Hamilton theorem. My professor noted that this should be proved. However, the proof of this fact is rather straightforward, no? Is the proof I ...
- 437
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votes
1
answer
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Cofactor of a tensor
In the book that I’m actually using for tensor algebra (second order tensors in $\mathbb{R}^3$), the author defines the cofactor of a tensor as the tensor that transforms the area vector, that is, ...
3
votes
1
answer
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Question regarding Proof of Cayley-Hamilton Theorem for Modules in Atiyah-Macdonald (Proposition 2.4)
Here is the proof of the Therorem:
Proposition 2.4. Let $M$ be a finitely generated $A$-module, let $α$ be an ideal of $A$, and let $\phi$ be an $A$-module endomorphism of $M$ such that $\phi(M) ⊆ αM$....
- 113
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1
answer
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Possibly wrong proof of Cayley-Hamilton Theorem
I have question on a proof idea I saw of Cayley-Hamilton theorem.
C-H Thm. A matrix $A \in M_n(\Bbb{R})$ satisfies its own characteristic polynomial.
The sketch of the proof is the following:
...
- 466
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votes
3
answers
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Let $n$ be a positive integer. Find $A^{4n}$ in terms of $n$.
I am presented with a matrix $A$
\begin{equation*}
A = \begin{bmatrix}
-8 & -30 & 10 \\
5 & 20 & -7 \\
10 & 33 & -10
\end{bmatrix}
\end{equation*}
and have been asked to find ...
- 481
2
votes
1
answer
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Efficient way to find power of matrix [duplicate]
Let
$$A=\begin{pmatrix}
2 & -1 & 2 \\
-1 & 2 & -1 \\
1 & -1 & 2
\end{pmatrix}$$
Find $B = A^{2021}-5A^{2020}+2A^{2019}+5A^{2018}-3A^{2017}+A^2$.
I tried eigenvalue ...
- 419
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votes
0
answers
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Trying to de-coordinate a proof of the Cayley-Hamilton theorem
There is a proof of the Cayley-Hamilton theorem which is presented in Introduction to Commutative Algebra by Atiyah and MacDonald, near the section on Nakayama's lemma. I am interested in how to turn ...
- 14.1k
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1
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views
How to prove an eigenvector belongs to a span of an unstable matrix?
Suppose we have a matrix $A\in\mathbb{R}^{n\times n}$ and a nonzero vector $p\in\mathbb{R}^{n}$.
If $\lim_{k\rightarrow\infty}A^kp\neq\textbf{0}$, then $A$ has an unstable eigenvector $v$ i.e., $Av=\...
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2
answers
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Show that $\exp(\alpha B)$ is represented by $I+(\sin\alpha)B+(1-\cos\alpha)B^2$ (Tokyo entrance exam [Math,2020])
I am currently trying to solve the following problem from the 2020 Tokyo entrance exam (math department):
第一問
正方行列A,Bおとすう。$$A=\begin{pmatrix}1&\sqrt2&0\\\sqrt2&1&\sqrt2\\0&\sqrt2&...
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votes
0
answers
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Image of the Squaring Function on $ \mathcal{M}_2(\mathbb{Q}) $
Find the image of the function
$
f : \mathcal{M}_2(\mathbb{Q}) \to \mathcal{M}_2(\mathbb{Q}), \quad f(X) = X^2, \quad \forall X \in \mathcal{M}_2(\mathbb{Q}).
$
I can’t find the correct approach for ...
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votes
1
answer
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Cayley Hamilton Theorem application explanation. [closed]
Question : Find the $n$-th power of the matrix $$\begin{bmatrix} 1 & 3 \\ -3 & -5 \end{bmatrix}$$
I found eigenvalues $\lambda = -2 ,-2$.
Then, I followed instructions as provided on this ...
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0
answers
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Confusion regarding Cayley-Hamilton theorem.
Theorem 4 (Cayley-Hamilton). Let $T$ be a linear operator on a finite dimensional vector space $V$ . If $f$ is the characteristic polynomial for $T$ , then $\mathrm{f}(\mathrm{T})=0$; in other words, ...
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How to prove that $\left\|A^d\right\|\leq C\rho(A)\left\|A\right\|^{d-1}$ using Cayley-Hamilton?
If $A$ is a complex matrix, and $\rho(A)$ is its spectral Radius, then Jairo Bochi's answer to this question,
$$\left\|A^d\right\|\leq C\rho(A)\left\|A\right\|^{d-1};\quad (Eq.\, I)$$
where $C>0$.
...
- 2,359
1
vote
1
answer
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Is "Since $f(x)\cdot I$ may also be regarded as a scalar matrix whose diagonal elements are the polynomials $f(x)$," necessary? (MacDuffee's book)
I am reading "Vectors and Matrices" by Cyrus Colton MacDuffee.
Let $R$ be a ring with unit element $1$. (We don't assume that $R$ is commutative.)
Let $f(x)=a_0+a_1x+\cdots+a_nx^n\in R[x]$.
...
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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 ...
- 21
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votes
3
answers
161
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Order of pole of $(zI-A)^{-1}$ at eigenvalue.
Let $A$ be a complex $n\times n$ matrix and $\lambda$ one of its eigenvalues. While reading this answer I was wondering what the order (denote it by $k_\lambda$) of the pole of $z\mapsto (zI-A)^{-1}$ ...
- 8,908
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Is this 1-line proof of Cayley–Hamilton incomplete?
In the comments of Martin Brandenburg's answer to this old MO question Victor Protsak offers the following "1-line proof" of the Cayley–Hamilton theorem. Here $p_A(\lambda)$ is the ...
- 477k
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Linear operator annihilates vector for every polynomial
I am working on this problem. I am studying for an exam.
Let $T$ be a linear operator on a finite-dimensional vector space $V$. Show that there is a non-zero vector $v \in V$ such that for all $f \in \...
- 186
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vote
2
answers
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views
Let $A, C \in M_{2}(\mathbb{R})$ such that $A = AC-CA$. Prove that $A^{2}=0$
Let $A, C \in M_{2}(\mathbb{R})$ such that $A = AC-CA$. Prove that $A^{2}=0$
My attempt.
Note that
$$
\operatorname{tr}(A) = \operatorname{tr}(AC)-\operatorname{tr}(CA) = \operatorname{tr}(AC)-\...
- 1,004
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votes
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answer
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Is this Corollary a historical or modern product?
I saw this Corollary on page 21 of this book "Introduction to commutative algebra" by Atiyah and Macdonald.
Corollary 2.5. Let $M$ be a finitely generated $A$-module and let $a$ be an ideal ...
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0
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How to understand a situation where one can use Nakayama's lemma even when the situation is not Tailor-made.
In commutative algebra we have the following version of Nakayama's Lemma(also calle NAK lemma):
NAK Lemma:
Let $R$ be a local ring and $\mathbf m$ be the unique maximal ideal of $R$.Let $M$ be a ...
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views
Given a $3\times 3$ $\operatorname{adj} A$, find $A$
Given $\operatorname{adj}A=\begin{bmatrix} -1 & -2 & 1\\ 3 & 0 & -3 \\ 1 & -4 & 1 \end{bmatrix}$ . Find $A$.
My Attempt
We know that $|\operatorname{adj}A|=|A|^{n-1}\Rightarrow ...
- 11.2k
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vote
1
answer
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How does the Cayley–Hamilton theorem give me a formula for eigenvectors
The wikipedia page Eigenvalues gives an example of how to compute eigenvectors of a matrix if already given the eigenvalues. The page claims this is an application of the Cayley–Hamilton theorem. ...
- 519
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votes
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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 ...
- 364
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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 ...
user1096489
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vote
3
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Let $A$ and $B$ be two square matrices of order $2\times 2$, where $\det(A)=1$ and $\det(B)=2$ then value of $\det(A+\alpha B)-\det(\alpha A+B)$
Let $A$ and $B$ be two square matrices of order $2\times 2$, where $\det(A)=1$ and $\det(B)=2$ then value of $$\det(A+\alpha B)-\det(\alpha A+B)$$ where $\alpha \in \mathbb{R}$ is
(A)$\alpha^2$
(B)$0$
...
- 11.2k
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$A$ is a square matrix of order $2$ with real entries and ${\rm Tr}(A)+|A|=2$. Show that $|A^2+|A|\cdot A+{\rm Tr}(A)I_2|\geq 4$
$A$ is a square matrix of order $2$ with real entries and ${\rm Tr}(A)+|A|=2$. Show that
$$|A^2+|A|\cdot A+{\rm Tr}(A)I_2|\geq 4$$
My Attempt
I could observe that ${\rm Tr}(A)+|A|=2\Rightarrow 1+a+d+...
- 11.2k
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vote
1
answer
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Cayley-Hamilton proof using linear discrete-time systems
So I had a question regarding proving the Cayley-Hamilton theorem using discrete states i.e.
$x(k+1)=Ax(k)+bu(k)$ & $y(k)=c^Tx(k)$ where $x(k),b,c \in R^n$.
The question stated that for an integer ...
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Showing there exists an $n\times n$ matrix that solves a given polynomial iff $n$ is even
Prove or disprove: There exists a real $n\times n$-matrix $A$ satisfying:
$$
A^2+2A+5I_n=0
$$
if and only if $n$ is even.
If $n=2$ this is quite easy, we can just compute the companion matrix. However,...
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0
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Calculation of the $3\times 3$ exponent matrix via Cayley-Hamilton theorem.
I have a random $3\times 3$ matrix $A$.
How can I calculate $e^A$ by $E$ (the identity matrix), $A$ and $A^2$, using the Cayley-Hamilton theorem?
I need a general expression that includes only the ...
- 11
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answer
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Show that $XY=0$ or $YX=0$
We have $X,Y$ $(2×2)$ matrices with complex entries and $X=A^{2}-B^{2}$ and $Y=AB-BA$. We know that $\det(X)=\det(Y)=0$. Show that $XY=0$ or $YX=0$.
I see that Trace of $Y$ is $0$ and $\det(Y)$ is ...
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vote
1
answer
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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 ...
- 1,331
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votes
1
answer
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Clarification of the details of the proof of Cayley Hamilton theorem in commutative algebra
I am trying to understand this proof of the Cayley Hamilton theorem from commutative algebra by Atiyah Mcdonald. So I am reading the following power point slides which gives more details but there is ...
- 4,751
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2
answers
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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}.$$
I ...
- 1,331
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Prove $\det((AB)^{n}-(BA)^{n})$ is a perfect cube.
We have $A,B$ two $3×3$ matrices with integer numbers. We know that $(AB)^{2}+BA=(BA)^2+AB$.
a) Show that $\det((AB)^{n}-(BA)^{n})$ is divisible by $det(AB-BA)$.
b) Show that if $\det(AB-BA)=1$, then $...
- 1,331
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votes
0
answers
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Theorem 4 (Cayley-Hamilton), Section 6.3 of Hoffman’s Linear Algebra
Let $T$ be a linear operator on a finite dimensional vector space $V$. If $f$ is the characteristic polynomial for $T$, then $f(T)=0$; in other words, the minimal polynomial divides the characteristic ...
- 4,645
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votes
1
answer
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Implication from definition of characteristic polynomial
I know that the characteristic function of a linear map $T:V\to V$ is defined as $\chi_T(x):=\chi_A(x)$ where $A$ is any matrix for $T$ (i.e. $Tv = Av$) with respect to some basis of $V$. I know this ...
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How to find the fourth power of a matrix just using Cayley Hamilton?
$$A=\begin{bmatrix}2&-1&2\\-1&2&-1\\1&-1&2\end{bmatrix}$$
I am supposed to find $A^4$.
I tried to find the characteristic equation and got
$$-x^3+6x^2-8x+3=0$$
From here I can ...
- 447
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vote
1
answer
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views
If $A$ commutes with $(AB - BA)^2$, is $\det(AB - BA) = 0$?
We have $A$ and $B$ are $3 \times 3$ matrices with complex numbers. We know matrix $A$ is commuting with matrix $(AB-BA)^2$. Can you show $\det(AB-BA)=0$?
I tried using some Hamilton Cayley Theorem on ...
- 1,331
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votes
1
answer
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Calculate matrix in negative power by using Cayley-Hamilton theorem
I have found the characteristic polynomial of a 2x2 matrix $A$:
$$λ^2-8λ+15=0$$
Through Cayley-Hamilton Theorem:
$$A^2-8A+15I=0$$
We are asked to calculate $A^{-2}$ as a function of $A$ and $I$.
I ...
- 81
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votes
0
answers
149
views
If Cayley-Hamilton holds over $\mathbb{C}$ it holds over all unitary, commutative rings.
I‘m currently learning about polynomial rings and am supposed to use them to show that: If the Cayley-Hamilton Theorem holdes over matrices over $\mathbb{C}$ then it holds for matrices with entries in ...
- 1,568
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votes
0
answers
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Step in proof of Cayley-Hamilton theorem in Steinberg's book
I am reading "Representation Theory of Finite Groups - An Introductory Approach" by Benjamin Steinberg, and making exercise 2.9. I can unfortunately not find a solution anywhere. Most of the ...
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votes
1
answer
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Is there any intuitive explanation to $A^2=\operatorname{tr}(A)A-\det(A)I_{2\times 2}$ for $A \in \mathbb{R}^{2\times 2}$?
$A^2=\operatorname{tr}(A)A-\det(A)I_{2\times 2}$ for $A \in \mathbb{R}^{2\times 2}$
This equation is easy to prove by denoting $$A = \begin{bmatrix}
a & b\\
c & d
\end{bmatrix}$$ but I am ...
- 23
2
votes
1
answer
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Representation of fields with matrices
I know that the ring
$(AS,+,\cdot)$, where
$$AS := \bigg\{\bigg( \begin{matrix} a & - b \\ b & a \end{matrix} \bigg) \; : \; a,b \in \mathbb{R} \bigg\}$$
and $+$ is the matrix addition and $\...
- 1,509
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votes
2
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Finding all polynomials $q,f$ such that $q(C)$ has non-zero kernel and $f(C)$ is invertible
This question comes from a qualifying exam.
Let $C$ be an $n × n$ real matrix with $n ≥ 3$.
(a) For which real polynomials $q$ of degree 2 is the null space of $q(C)$ not the zero subspace?
(b) For ...
- 3,065
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votes
2
answers
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Algebraic Elements are Integral, if their Minimal Polynomial is.
In an upcoming exercise class in commutative algebra I would like to discuss how to detect, whether an algebraic element $\alpha$ over $\Bbb Q$ is integral over $\Bbb Z$. The claim is that it is ...
- 12.2k
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votes
0
answers
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Vandermonde matrix whose elements are different roots of unity
I am solving the following linear system:
$$c_q=\sum_ka_{k,q} f_k\\ a_{k,q}=\exp\left({2 \pi i\frac{k}{q+1}}\right)$$
with $ 0\leq k\leq m,0\leq q\leq m$. For this, it would be useful to calculate its ...
1
vote
2
answers
76
views
Understanding properties of a matrix $A\in \mathcal{M}_n({K})$ for which $C(A)=\{f(A): f(x) \in K[x]\}$.
Consider the set of all square matrices with $n$ columns over $K(\Bbb{R} \text{ or} \Bbb{C})$: $\mathcal{M}_n({K}) $
Define $Z(\mathcal{M}_n({K})) = \{A\in \mathcal{M}_n({K}) : AB=BA, \forall B \in \...
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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