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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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Cayley Hamilton Theorem Intuition

Why should, intuitive (not a formal proof, just motivations ) be true that the square matrix satisfy its own characteristic equation?
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Find the characteristic and minimal polynom of the following transformation

The transformation is $$T:M_{\leq n}(\mathbb{C})\rightarrow M_{\leq n}(\mathbb{C})$$ and defined by $$T(A) = A^t-A.$$ If we try to take a basis $B$ and calculate the determinant of $xI-[T]_B$, we get ...
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Finding $A^{-3}$ using Cayley Hamilton Theorem

If $$A = \begin{bmatrix} 2 & 4 \\ 1 & 1 \\ \end{bmatrix}, $$ then use the Cayley-Hamilton Theorem to find $A^{-3}$. This is how far I have gotten: \begin{align} p(\lambda) &= \lambda^2 -...
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Questions on the proof of Cayley Hamilton Theorem

In the proof presented in my textbook, it utilises the equation $$(A-\lambda I) \operatorname{Adj}(A - \lambda I) = \det(A-\lambda I) \, I$$ And it stated that, $$P(A-\lambda I) = \det(A-\lambda I)=...
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Does the Cayley–Hamilton theorem work in the opposite direction?

The Cayley–Hamilton theorem states that every square matrix satisfies its own characteristic equation. But does it work in the opposite direction? If for example for a certain matrix $A$ we know ...
2
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1answer
147 views

Functorial proof of Cayley-Hamilton using exterior powers

Let $V$ be a rank $n$ free module over a commutative ring $R$. Let $\dagger$ denote the adjoint with respect to the natural perfect pairing given by the wedge product $$\textstyle \bigwedge^k\otimes \...
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1answer
87 views

Cayley-Hamilton says that evaluating an endomorphism's characteristic polynomial over that endomorphism gives zero. Isn't this true by substitution?

Given an endomorphism $f$ of a vector space $V$, its characteristic polynomial, say $P(x)$, is defined as follows: $P(x) = \det(f -xI)$, where $I$ is the identity endomorphism. It is well known that, ...
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Show that $e^{tA} = \sum\limits_{k=0}^{n-1} f_k(t)A^k$

Let $A$ be a $n\times n$ matrix such that the characteristic polynomial of $A$ is $$P(\lambda)=\lambda^n+a_{n-1}\lambda^{n-1}+...+a_1\lambda+a_0$$ Now consider the nth order differential equation $$\...
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How to prove Cayley Hamilton theorem? [duplicate]

Today I read about a new theorem in matrix that is the Cayley Hamilton theorem which has a mathematical expression as $|A-LI|=0$. Can anyone tell me how was it derived .I know that the expression is ...
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2answers
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Finding a cubic polynomial with Cayley-Hamilton Theorem

I have two matrices: $A = \begin{bmatrix} 1 &2 \\ -1 &4 \end{bmatrix} $ and $B = \begin{bmatrix} 0 &-1 \\ 2 &3 \end{bmatrix} $ I need to find a monic cubic polynomial $g$ such ...
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Inverse function on matrices with help of Cayley-Hamilton theorem?

I have been thinking about inverse functions of matrices lately. (Yees yees, I know usually for anything more complicated than reals we need to define/select branch and for reals to select interval ...
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1answer
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Does Cayley Hamilton Theorem apply for non-diagonalizable matrices as well?

Cayley Hamilton Theorem says that a matrix $A$ satisfies its characteristic equation. My professor proved this for diagonalizable matrices. What happens if $A$ is not diagonalizable? Does the C-H ...
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Motivation - Proving Cayley-Hamilton with Adjoints

The first proof on Wikipedia on Cayley-Hamilton (a direct algebraic proof) goes on about proving the theorem by considering the adjoint to have some comparison for the characteristic polynomial. Is ...
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Can the span of $L(I,A,A^2,\cdots,A^k,..)$ have more than $n$ elements?

Let $A\in M_n$. Can dimension of subspace $L(I,A,A^2,\ldots,A^k,\ldots)$ of $M_n$ can be bigger than $n$? Using Cayley Hamilton theorem that every matrices $A^n$ can expressed as linear combination ...
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Vector subspace of $\mathbb{C}$ spanned by $n$ complex numbers over the field of rational numbers.

Let $\alpha_{1} \dots \alpha_{n}$ be complex numbers and V = $\{ \sum_{i=1}^{n} a_{i}\alpha_{i} : a_{i} \in \mathbb{Q} \}$ be the vector subspace of $\mathbb{C}$ spanned by them over the field of ...
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Given a $3\times3$ real matrix $A$ with eigenvalues $0,1,2$, find real constants $a,b,c$ such that $aI+bA+cA^2$ has eigenvalues 0,1,3

Suppose $A$ is a $3\times 3$ real matrix with three distinct eigenvalues $0,1,2$. Find real constants $a,b,c$ such that the matrix $aI+bA+cA^2$ has eigenvalues $0,1,3$. My only initial thought on how ...
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Cayley–Hamilton And Invertible Matrix

In my lecture notes, it was mentioned that if the Cayley–Hamilton polynomial has a free element then it is invertible. Namely, $P_A(x) = a_n x^n + \dots + a_1 x + a_0$ there $a_0 \neq 0$. Why is it ...
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Putzers Algorithm to compute $A^n$.

I'm reading through An Introduction to Difference Equations, Saber Elaydi, Third Edition. And I have come to a part i can't quite figure out, it's about computing $A^n$ by Putzers Algorithm. The ...
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1answer
243 views

Why Hamiltonian cycle decision problem in NP-complete?

I was reading the algorithm book of Neapollian and Naeempoor and it says Hamiltonian cycle decision problem is np complete and CNF can be reduced to it .I understand that why it is NP but i want to ...
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Statement about Generalised Eigenvectors Involving Power of Algebraic Multiplicty

Reading a proof for the Cayley-Hamilton Theorem, I came across the following statement, If $\textbf{v}$ is a generalised eigenvector of an $n \times n$ matrix $A$ with corresponding eigenvalue $\...
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1answer
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Can I use the Cayley-Hamilton theorem to determine $ \det(A) $ from the identity $ {A}^{-1} = {A}^{2} - c {I}_{n} $?

The exercise is the following: Assume that $A$ is a $n \times n$ matrix satisfying the identity $A^3-cA=I_n$ ($c>0$). a) Prove that $A^{-1}=A^2-cI_n$. b) Compute the possible values of $\det(A)$. ...
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Which one of the following are correct?

Let, $A= \left[ {\begin{array}{cc} -1 & 2 \\ 0 & -1 \\ \end{array} } \right]$ , and $B = A + A^2 + A^3 +···+ A^{50}$. Then $(A) B^2 = I $ $(B) B^2 = 0$ $(C) B^2 = A$ ...
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1answer
25 views

Dimension on span of End(V) composition

Supose that W in a vector space with finite dimension. Let $f \in End(V)$, also $f^n \in End(V)$ for $ n $ in positive natural numbers. Where $f^0=Id$, and for $n\ge1$, $f=f^{n-1}\circ f$, $f^2=f$. ...
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1answer
27 views

Prove that characteristic polynomial = minimal polynomial

Let $M \in M_{3}(K)$ whereby $K$ is a field, such that $\chi_{M}$ only has one root in $K$. Show: $\chi_{M}=\mu_{M}$ whereby $\mu_{M}$ is the minimal polynomial. Steps I thought about: Let $\...
2
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1answer
54 views

Write $m_{I-A}$ (the minimal polynomial of $I-A$) in terms of $m_A$ and conclude: $\deg(m_A)=\deg(m_{I-A})$

Given a matrix $A$ and its minimal polynomial $m_A$. Write $m_{I-A}$ (the minimal polynomial of $I-A$) in terms of $m_A$ and conclude: $\deg(m_A)=\deg(m_{I-A})$ Any hint or direction would be ...
3
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3answers
77 views

$A^2=-I_4$. Find possible values of minimal polynomial and characteristic polynomial

Let $A\in\mathbb{R}^{4\times 4}$ satisfy $$A^2=-I_4 .$$ (a) Find possible values of $m_a$ (minimal polynomial) and $p_a$ (characteristic polynomial). (b) Find an example for A satisfying the ...
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For $2 \times 2$ matrices $AB=-BA$ with $BA$ not $0$, prove that $\mathrm{tr}(A)=\mathrm{tr}(B)=\mathrm{tr}(AB)=0$

It is easy to derive from $AB=-BA$ that $\mathrm{tr}(AB)=0$ since $\mathrm{tr}(AB)=\mathrm{tr}(-BA)=-\mathrm{tr}(BA)=-\mathrm{tr}(AB)$. However, I cannot get that $\mathrm{tr}(A)=\mathrm{tr}(B)=0$ ...
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1answer
71 views

$ X^{25} = A $ in $ M_{2}(\mathbb{R}) $

Let $ A = \bigl(\begin{smallmatrix} 2& -1\\ 4& -2 \end{smallmatrix}\bigr) $ Find the number of solutions from $ M_{2}(\mathbb{R}) $ of the ecuation $ X^{25} = A $ Since $ A^2=O_{2} $ and ...
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1answer
51 views

Finding The Characteristic Polynomial of a Matrix With Integer Sum Coefficients.

Good day, I'm having a bit of trouble with this one, I'm given the matrix \begin{bmatrix}1&2&...&n\\n+1&n+2&... &2n\\...&...&...&...\\n^2-n+1&n^2-n+2&...&...
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1answer
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How to prove a matrix is invertible given a polynomial that annihilates it?

The question reads: Let $A\in M_{10}$, let $f(z)=z^{4}+11z^{3}-7z^{2}+5z+3$, and suppose $f(A)=0$. Prove that $A$ is invertible and find a polynomial $g$ of degree $3$ or less such that $A^{-1}=g(...
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Show that $v$ is a eigenvector of $q(A)$ with respect to $p(\lambda)$.

Let $K$ a field, $p=\sum_{i=0}^k \alpha_i t^i \in K[t]$ and $A\in K^{n,n}, \quad n\in \mathbb N\setminus\{0\}$. Define $q:K^{n,n}\to K^{n,n}, q(A)=\sum_{i=0}^n \alpha_i A^i$. Let $\lambda \in K$ a ...
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Mistake in Peter Lax Linear Algebra Caley Hamilton proof

This is with reference to the Proof of Cayley Hamilton theorem on Page 68 of Peter Lax Linear algebra and its application (second edition) Equation 21 reads According to formula 30 of Chapter 5 $$...
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Cayley-Hamilton theorem to compute this.

$ A = \pmatrix{0&-3&0\\3&0&0\\0&0&-1}$ Compute the $e^{At}$. Well, the first problem of this is to calculate the inverse of $A$ using Cayley-Hamilton theorem. But for this ...
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Clarification on Cayley-Hamilton theorem [duplicate]

Cayley-Hamilton says that a square matrix $A$ over any commutative ring $R$ satisfies its own characteristic equation. Conversely, suppose that $p(T) \in R[T]$ is satisfied by $A$, i.e., that $p(A) ...
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2answers
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Let $A$ be a $(n \times n)$ complex matrix, and for $C$ a $(1 \times n)$ row vector let $W = \{\sum_{j=0}^{n-1}a_jCA^j:j=0,\dots,n-1\}\;.$

(i) Use the Cayley-Hamilton theorem to show that right multiplication $X \mapsto XA$ for $X \in W$ defines a linear operator from $W$ to $W$. I'm really not sure to answer this question. I know ...
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1answer
58 views

Structure of product of matrices and Cayley-Hamilton

Let $A \in \mathbb{R}^{n \times n}$ be a square $n$-dimensional real valued matrix. Let $X \in \mathbb{R}^{a \times n}$ and $Y \in \mathbb{R}^{n \times b}$. Define $M_{n-1}=XA^{n-1}Y$ and $M_{n}=XA^nY$...
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1answer
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Proof of Cayley-Hamilton Theorem in infinite fields only?

While trying to prove the Cayley-Hamilton theorem, I came up with the following proof: If $A$ is a diagonalizable matrix, so $A=SDS^{-1}$ with $D$ diagonal, then, letting $$P(\lambda)=\det(A-\lambda ...
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169 views

Evaluating Determinant using Cayley-Hamilton Theorem

A is a non singular square matrix of order 2 such that |A + |A|adjA| = 0, where adjA represents adjoint of matrix A, and |A| represents det(A) (determinant of matrix A) Evaluate |A – |A|adjA|. I ...
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Show that there is a polynomial $p(z)$ of degree at most $k$ such that $e^A = p(A)$.

Let $\dim V<\infty$, and let $A\in L(V)$ be diagonalizable with distinct eigenvalues $\lambda_0, \lambda_1,\ldots,\lambda_k \in \mathbb{N}_0$. Show that there is a polynomial $p(z)$ of degree at ...
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1answer
19 views

Condition of containing the eigenvectors of a Linear Operator over a finite dimensional vector space in a T invariant Subspace

Let $T$ be a linear operator on a finite dimensional vector space $V$ and $W$ be a $T$-invariant subspace of $V$.Suppose $v_1,v_2,...,v_n$ are the eigenvectors corresponding to distinct eigenvalues $\...
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2answers
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What is a max non hamiltonian graph?

Assume we're taking about simple graphs here. What is the exact definition of a non-hamiltonian graph ? Is it true if I say that: Adding one single edge to the max non-hamiltonian graph, will make ...
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1answer
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How to find $A^{100}$ by using the Cayley Hamilton theorem? [closed]

How to find $A^{100}$ as a linear function of A by use of the Cayley Hamilton theorem? The characteristic equation leads to $A^2-4A+3I=0$. I can't find any way to solve this.
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1answer
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Minimal polynomial substitution

This is probably a very easy question, I'm just not strong at the concept. Let $ A $ be an $ n \times n $ matrix with coefficient in a field $ F $. Then $ A $ determines a linear map $ T: F^n \to F^...
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255 views

Determinant Trick (Unfaithful module)

In the proof of one version of Nakayama's Lemma, one might use the Determinant Trick which is the following result. Let $R$ be a commutative ring with $1$. Let $I$ be an ideal of $R$ and $M$ be a $R$-...
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1answer
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Cayley Hamilton use to find higher power of non diagnalizable matrix. Reduction not easy for the characteristic equation. [closed]

Mat $A= \left(\begin{smallmatrix}1&0&0\\ 1&0&1\\ 0&1&0\end{smallmatrix}\right)$ Find $A^{30}$. Cannot diagonalize. Not generic reduction of characteristic ...
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1answer
75 views

If $A$ and $B$ have characteristic polynomial $x^5-x^3$ and minimal polynomial $x^4-x^2$, then they're similar.

Let $A$ and $B$ be matrices with coefficients in some field $k$. If they both have characteristic polynomial $x^5-x^3$ and minimal polynomial $x^4-x^2$, then they must be similar. I've thought about ...
2
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1answer
152 views

Find all $4 \times 4$ real matrices such that $A^3=I$

Find all $4 \times 4$ real matrices such that $A^3=I$. The minimal polynomial must divide $x^3-1$. Since the matrix is real, the minimal polynomial must be either $x-1$ or $x^3-1$ (i.e., if it ...
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1answer
136 views

Use Cayley Hamilton theorem to derive the following $(sI−A)^{-1}$ where $s \in \mathbb C$ .

$(sI−A)^{-1}$ where $s\in\mathbb C$. My attempt : I have seen this method of matrix equation using the Cayley Hamilton method . $f(A)= (sI−A)^{-1} = h_0(I) + h_1(A)+ h_2(A)^2$; $f(p) = h(p) = h_0 ...
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1answer
133 views

Using Cayley-Hamilton to calculate powers of arbitrary matrices of fixed size

For a symmetric matrix $A$ with eigenvalues $\lambda_i<1$ (with $i=0,\dots,n$), one can write the resolvent ($R(z,A):=(A-zI)^{-1}$) at $z=1$ as a series of powers of $A$ $[1]$: $R(1,A)=(A-1)^{-1}=-...
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3answers
110 views

Prove that all the powers of these two matrices have the same trace

Let $A, B \in M_{2}(\mathbb{R})$ be two square matrices such that $$AB \ne BA$$ and $$A^3 = B^3$$ Prove that $\mbox{Tr} (A^n) = \mbox{Tr} (B^n)$ for all $n \in \mathbb{N}$, where $\mbox{Tr} (\cdot)$ ...