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.

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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 ...
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Cayley-Hamilton Theorem explanation. I'm having trouble interpreting the meaning of $f(T)$.

Using the Theorem from "Linear Algebra" (-Friedberg, Insel, Spence 5th edition). (Let $T$ be a linear operator on a finite-dimensional vector space $V$, and let $f(t)$ be the characteristic ...
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How to express det (a(A-B) - bI)

I was looking over the solution of a problem and don't understand how they expanded the determinant. The problem: Let $A$, $B$ be square $2 \times 2$ real matrices, such that $(A-B)^2 =O$ (where $O$ ...
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Finding $\det(f(A))$, given the eigenvalues of $A$.

Consider a matrix $A$ with given eigenvalues. Given any expression involving $A$ and its inverse as $f(A)$. If I wish to find $\det(f(A))$, is there any algorithmic approach that may be followed to ...
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Eigenvalue of a $3\times 3$ complex matrix

In the following question I got that $0$ is not an eigenvalue, then $M$ is invertible and using Cayley-Hamilton then I got the last option correctly, then why given that $\alpha+\beta \neq 0$? What is ...
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Inverse of the Cayley-Hamilton theorem

Consider a $n\times n$ matrix A, say over $\mathbb{R}$ and $t \in \mathbb{R}$. In this Wikipedia article we read that Recall from above that an $n×n$ matrix $\exp(tA)$ amounts to a linear combination ...
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Interesting question about multiplication with matrix whose elements are from $L(f)$

$h=a_0+a_1x+..+a_nx^n$ $h(f)=a_01_V+a_1f+..+a_nf^n$ $f^n=f\circ f^{n-1}$ $\beta_f:R[X]\to L(V)$ ($L(V)=f:V\to V|f$ is linear) $Im(\beta_f)=L(f)$ Now we want to define multiplication of $(e_1,..,e_n)\...
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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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About a proof of Cayley-Hamilton theorem in "Linear Algebra" by Ichiro Satake.

I am reading "Linear Algebra" by Ichiro Satake. The author proved that $B_0,B_1,\dots,B_{n-1}$ commute with $A$. But I think we don't need to prove that $B_0,B_1,\dots,B_{n-1}$ commute with $...
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Proofs of the Cayley-Hamilton Theorem [duplicate]

The idea of this post is for people to post different proofs of the Cayley-Hamilton Theorem. You can either try to post your own proof or give a reference. If you usse a reference, please give some ...
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Prove Cayley-Hamilton theorem

I have a exercise in my linear algebra textbook: Let $c_2\lambda^2+c_1\lambda +c_0=0$ be the characteristic equation for the matrix $$A=\begin{pmatrix}1&3\\3&1\end{pmatrix}$$ Prove that $c_2A^...
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Find $5\times5$ invertible matrix $A$ over $\mathbb{F}_3$ such that $A^{-1} = 2A^3 +2I$, $A \neq I$ [closed]

I have tried to solve the above using the cayley hamilton theorem which yields nothing as I get $-1$ which is not in my field. I feel like I need to do a sub-block decomposition.
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Why can we substitute matrix and eigenvalues into other than the characteristic polynomial (Cayley Hamilton)

Generally speaking, this is a question about when we can swap $A \rightarrow\lambda$. In problem 2-5(c) of Applied Optimal Estimation, we are asked to consider the matrix $$A = \left[\begin{matrix} 1 &...
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Cayley-Hamilton in Macaulay2

The question arose in a more complicated situation - but this simple example will illustrate my difficulty. Suppose I want to use Macaulay2 to check the Cayley-Hamilton theorem for matrices of some ...
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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 ...
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Proving Cayley Hamilton theorem in a different way

I'm looking into the proof of Cayley Hamilton theorem from multiple sources. All the proofs I saw use the fact that $$ (A - \lambda I ) \,\mbox{adj} (A - \lambda I ) = | A - \lambda I | I $$ The ...
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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 ...
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Is the Putzer algorithm markedly more efficient than the Cayley-Hamilton reduction of $\exp(At)$?

The Putzer algorithm is an interesting and very remarkable (to me) simplification of the problem of taking $\exp(At)$ for any $t$. However, it does have some complex precomputation steps; that is, ...
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4 answers
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Find trace and determinant of matrix $A$ such that $A^2 = I$.

I have a $ 2\times 2$ matrix $A$, where $A^2 = I$. So the eigenvalues are $\lambda= \pm1$ . I need to find its trace and determinant. There's no mention of upper or lower triangular matrix, therefore,...
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5 votes
1 answer
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Can we determine higher powers of a matrix in terms of lower powered matrices?

Consider a n-ordered square matrix A. Using Cayley-Hamilton Theorem, I can represent the matrix $A^n$ as a matrix polynomial P(A) of degree n-1. Further any matrix $A^k$ where $k>n$ can also be ...
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Using Cayley-Hamilton to prove the following question

I recieved the following question: Let $A \in \mathbb M_k(\mathbb R)$ and let $U=span \lbrace I,A,A^2,A^3,.. \rbrace \subseteq \mathbb M_k(\mathbb R) $. Let $p(x)=a_0+a_1x+...+a_nx^n$ where $a_n \neq ...
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2 votes
3 answers
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Why Nilpotent Matrix is not Null matrix always?

I know it might sound dumb, but specifically, Why NULL matrices are not the only NULPOTENT matrices? I am thinking that as all eigen values of NILPOTENT matrices are 0, then $\lambda = 0$, and as per ...
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Hamilton-Cayley Theorem for Vector Spaces Over Any Field

In Serge Lang’s third edition of Linear Algebra on p. 243, he gives the result of the Hamilton-Cayley theorem for linear operators for vector spaces over any field $K$. My problem is that I don’t ...
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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 ...
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6 votes
2 answers
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Proof of Cayley-Hamilton using Krylov subspaces

I came up with another proof of the Cayley-Hamilton Theorem. Is this new? The proof is by induction over the dimension of the underlying vector space. Let $v \in \mathbb F^n \setminus \{0\}$. ...
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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&...
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1 answer
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Matrix roots of the Hamilton–Cayley equation

The Hamilton–Cayley theorem, or Cayley–Hamilton theorem, says that every $n\times n$ matrix is a zero of its own characteristic polynomial. The ring of $n\times n$ matrices is not a field and in ...
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1 answer
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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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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 ...
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1 answer
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Different values of $A^n$ using Cayley-Hamilton Theorem And Direct Multiplication

Let's say there's a matrix A $$ A = \begin{pmatrix} 3 & -4\\ 1 & -1\\ \end{pmatrix} $$ Now I want to find $A^n$ I tried the following two methods but get different answers. Any ...
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0 answers
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Questions about the Cayley-Hamilton theorem for modules

Having recently learned the proof of CH for vector spaces from Hoffman&Kunze (I've known the statement of the theorem for a while now, but have never really bothered with the proof), I am now ...
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2 votes
1 answer
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Representing matrix exponential in terms of matrix powers via Cayley-Hamilton

Using the Cayley-Hamilton theorem, show that for a $2 \times 2$ matrix $A$, $$e^A = c_1 A + c_0 I$$ where $c_1$ and $c_2$ are constants. This problem came to me absolutely from nowhere and I have no ...
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2 votes
2 answers
76 views

Finding eigenvalue of a matrix expression

Consider the following matrix $A=\left[\begin{array}{lll}1 & 0 & 0 \\ 2 & 3 & 0 \\ -3 & 1 & -2\end{array}\right]$ How can I find the eigenvalues of $3 \mathrm{~A}^{3}+5 \mathrm{...
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Cayley-Hamilton theorem and $\prod_{i=1}^nA-\lambda_iI$ ($\lambda_i$ is the $i$th eigenvalue)

How does Cayley Hamilton theorem link to $\prod_{i=1}^nA-\lambda_iI$? All definitions of the theorem I can find say that $p(A)=0$ (the characteristic polynomial of $A$ is $0$). Thanks!
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What does it mean geometrically for a matrix to satisfy its characteristic equation or to satisfy the Cayley-Hamiltonian theorem?

I am a high schooler who has just recently learned a bit about matrices and the Cayley-Hamiltonian theorem. And my teachers have told me that to get the characteristic equation of many matrix $A$, I ...
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2 votes
3 answers
131 views

Using the converse if Cayley Hamilton theorem

My textbook says : Let $M$ be a $3 \times 3$ Hermitian matrix which satisfies the matrix equation $$ M^{2}-5 M+6 I=0 $$ Where $I$ refers to the identity matrix. Which of the following are possible ...
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2 votes
1 answer
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Is there a way to prove the Cayley-Hamilton Theorem without the use of cofactors, adjoints, etc?

Is there a way to prove the Cayley-Hamilton Theorem without the use of cofactors, adjoints, etc? Like is there another way to natural prove general matrix will satisfy its own characteristic ...
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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 ...
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1 vote
1 answer
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Help with Cayley-Hamilton determinant trick theorem from Matsumura's Commutative Algebra.

Theorem 2.1. Suppose that $M$ is an $A$-module generated by $n$ elements, and that $\varphi \in \text{Hom}_A(M,M)$; let $I$ be an ideal of $A$ such that $\varphi(M) \subset IM$. Then there is a ...
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2 votes
1 answer
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What is going on in this proof of the Cayley-Hamilton theorem?

I am reading a proof of the Cayley-Hamilton theorem here. For a rough outline of the proof, let $A$ be a matrix representing the endomorphism $\phi$ over finitely-generated module $M$ with generators ...
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2 votes
0 answers
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Cayley Hamilton proof [duplicate]

I'm currently studying Cayley-Hamilton proof, but I think I'm missing something massive. The theorem states that an endomorphism $\phi$ cancels its characteristic polynomial. I don't understand why ...
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1 answer
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If $ A $ is a $ 2 \times 2 $ real matrix such that $ \det (A) = 1 $ and $ A^n = I$ show that $ A ^tA = I $

If $ A $ is a $ 2 \times 2 $ real matrix such that $\det (A) = 1 $ and $ A^n = I$ show that $ A ^tA = I $ IDEA: Since $ \text{det}(A) = 1 $ according to the Cayley-Hamilton theorem, it is true that $$...
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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 $(A+I_3)^{10}(A-I_3)^2+A$...
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1 vote
1 answer
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Extension of the field of a matrix vector space

A pretty common step that I'm encountering in linear algebra proofs is "extending the field" in which a matrix is defined. I'll make some examples: Cayley-Hamilton theorem The book I'm using ...
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A question in Corollary Section 7.1 of Hoffman Kunze Linear Algebra

I am self studying Chapter -7 of Linear Algebra from Hoffman Kunze and I have a question in 1st section in last corollary whose image I am adding . Image of Theorem 1: I have a question in 1st ...
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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 ...
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1 vote
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Cayley-Hamilton-Theorem - Possible characteristic polynomial

Let $A: \mathbb{R}^3 \to \mathbb{R}^3$ s.t. $A^3-2A^2+A= 0$ The Cayley-Hamilton-Thm. states that if I put $A$ into its characteristic polynomial it'll equal $0$. But am I allowed to conclude from the ...
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3 answers
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On the Cayley-Hamilton theorem

One of the nicest theorems in linear algebra is the one that a matrix satisfies its own characteristic polynomial, the so-called Cayley-Hamilton theorem. What is a good way to prove it? In particular,...
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1 answer
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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: $$\left(x-1\right)^3\left(x-2\right)^4$$ $$\dim\:\ker\:\left(A-2I\right)=3$$ $$\...
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  • 1,258
2 votes
1 answer
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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 ...
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