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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59 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 $(A+I_3)^{10}(A-I_3)^2+A$...
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Proof of The Cayley Hamilton theorem and special cases [closed]

Can you please provide me the proof of The Cayley Hamilton Theorem and why it would not give identity matrix as a root of a characteristic equation even if the unit matrix satisfied the equation.
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1answer
21 views

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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1answer
27 views

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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1answer
84 views

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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18 views

Certain endomorphisms of a finitely generated module satisfies polynomial (proposition 2.4 in Atiyah McDonald)

I am trying to understand the proof of the following proposition, Let $M$ be a finitely generated $A$-module, and $\frak{a}$ be an ideal of $A$. Let $\phi$ be an endomorphism of $M$ such that $\phi(M) ...
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1answer
28 views

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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1answer
78 views

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 "the" proof. I am hopeful that it is ...
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1answer
34 views

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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1answer
26 views

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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44 views

Cayley-Hamilton theorem for symmetric positive definite matrix

Let $A$ be a symmetric positive definite matrix. We know that by Cayley-Hamilton theorem \begin{aligned}A^{-1}={\frac {(-1)^{n-1}}{\det A}}(A^{n-1}+c_{n-1}A^{n-2}+\cdots +c_{1}I_{n}).\end{aligned} ...
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2answers
27 views

Does a square matrix satisfy only its characteristic equation?

Say that we have a square matrix of order $N \times N$ and is NOT a diagonal matrix. Now can this matrix satisfy a polynomial of degree $N$ other than its own characteristic equation?
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38 views

Let A be an $n \times n$ matrix. Prove that $dim(span{I_n,A,A^2,…}) \leq n$. [duplicate]

Let A be an $n \times n$ matrix. Prove that $dim(span{I_n,A,A^2,...}) \leq n$. I am thinking about the theorem( Let T be a linear operator on a finite-dimensional vector space V, and let W denote the ...
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1answer
44 views

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 ...
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1answer
35 views

Inverse of a symmetric using Cayley-Hamilton theorem

Let $A$ be a symmetric positive definite matrix. I am interested in finding a general expression for the $A^{-1}$. Using the Cayley-Hamilton Theorem (https://en.wikipedia.org/wiki/Cayley%E2%80%...
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37 views

Where does the distinction between $det(\lambda I - A)$ and $det(A - \lambda I)$ for char. poly. of A come from?

I see certain textbooks and people use one or the other and i did the arithmetic on one very basic case and they happenned to come out the same, but I'm still curious where does the difference ...
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1answer
65 views

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 ...
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1answer
56 views

Corollary 2.5 from Atiyah-Macdonald small question

I'm having a doubt in the proof of Corollary 2.5 from Atiyah-Macdonald's Introduction to Commutative Algebra (page 21). The proof is simple: apply Proposition 2.4 (basically a version of Cayley-...
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1answer
67 views

6×6 Matrix Exponential using the Cayley-Hamilton theorem

For the $6×6$ matrix $$P= \begin{pmatrix} 0 & 0 & 0 & 0 & 2 & 0 \\ -1& 0 & 0 & 4 & 2 & 0 \\ 0 & 1 &-1 & 0 &...
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1answer
94 views

Can we find / express all matrices so that ${\bf P}^2 = {\bf P+I}$?

Can we find/express all matrices, so that: $${\bf P}^2={\bf P+I}$$ Own work: For the eigenvalues must then hold: $$\lambda^2-\lambda-1=0$$ In other words : $$\lambda_1 = \frac{1-\sqrt{5}}2\\\...
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3answers
132 views

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 \\...
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4answers
117 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 $...
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1answer
46 views

Direct sum of subspaces using Cayley-Hamilton

$\renewcommand{\phi}{\varphi}\newcommand{\id}{\text{id}}$Consider an endomorphism $\phi:V\to V$, where $V$ is any finite dimensional $F$-vector space. Assume the characteristic polynomial $\chi_\phi=...
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2answers
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Let $A$ be a real $2×2$ matrix such that $A^6=I$. The total number of possibilities for the characteristic polynomial of $A$ is:

Let $A$ be a real $2×2$ matrix such that $A^6 = I$ (where $I$ denote the identity $2×2$ matrix). The total number of possibilities for the characteristic polynomial of $A$ is: Annihilating polynomial ...
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34 views

Prove using Cayley-Hamilton

If $A$ is a symmetric $n{\times}n$ matrix with $n$ distinct eigenvalues $λ_i$ show that any polynomial $P(A)$ can be expressed in the form: $$ P(A) = c_1A^{n-1} + c_2A^{n-2} +\dots+ c_{n-1}A + c_n I, ...
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1answer
46 views

Find a 4 x 4 matrix with rational entries such that its fourth power is the identity matrix multiplied by -1

I am working on the following problem: Find a $4 \times 4$ matrix over $\mathbb{Q}$ such that $A^4 = -I$. I know that if $A^4 = -I$, it would suffice to find a $4 \times 4$ matrix over $\mathbb{Q}$...
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48 views

Solving exponential matrix using Cayley–Hamilton theorem, I got stuck.

Using the Cayley–Hamilton theorem, I got the following matrix exponential (for $3 \times 3$ matrix $A$): $$e^{At}=\left(e^t - te^t+\dfrac{1}{2}\, t^2e^t\right) E+\left(te^t-t^2e^t\right)A+\dfrac{1}{...
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1answer
43 views

Range of Controllability is A-Invariant

Consider the controllability matrix $C= [B, AB, A^2B,...,A^{n-1}B]$ such that $A \in \mathbb{R}^{n\times n}$ and $B \in \mathbb{R}^{n\times m}$. Now consider the range of $C$, $\mathcal{R}(C)= \{v | ...
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42 views

Finding $n$-th power of a $2 \times 2$ matrix with $2$ identical eigenvalues

If$$ A = \begin{pmatrix} 3 & -4 \\ 1 & -1 \\ \end{pmatrix} $$ prove that $$A^k = \begin{pmatrix} 1+2k & -4k \\ k & 1-2k \\ \end{pmatrix}$$ Now the first ...
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1answer
33 views

Does every polynomial divide its (Galois-theoretic) norm?

In Hermann Weyl's The Algebraic Theory of Numbers (1940 [reprinted 1951]), section 1.9 is a discussion of change of base-field (I believe).  I'm having a bit of trouble following. Let $\kappa$ ...
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2answers
102 views

Let $A=\begin{bmatrix} 1 & 2\\ 3& 4 \end{bmatrix}$ then det$(A^3-6A^2+5A+3I)=3$

Let $A=\begin{bmatrix} 1 & 2\\ 3& 4 \end{bmatrix}$ then det$(A^3-6A^2+5A+3I)=3$ det$(A^3-6A^2+5A+3I)=$det$((A^2-5A-2I)(A-I)+2A+I)= $det$(2A+I)=3$, Since a matrix satisfies its ...
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24 views

Given a matricial expression, the eigenvalues are restricted to the polynomial solutions?

Given a matricial expression, like $A^2-4I=0$, I want to know if it is true that the eigenvalues of $A$ are restricted to the solutions of the polynomial $x^2-4=0$, so $x \in \{-2,2\}$. With Cayley-...
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3answers
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Proofs Using Upper Triangular Form Over $\mathbb{C}$

I recently came across a proof for Cayley-Hamilton for any upper triangular matrix and then generalizing to all square matrices over $\mathbb{C}$. There was no explanation as to why we can generalize ...
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Do we really need the eigenvectors of $A$ to find $e^{tA}$?

For an $n\times n$ matrix $A$, using the Cayley-Hamilton theorem, we can express $$e^{tA} = \alpha_1(t)I + \alpha_2(t)A + \cdots + \alpha_n(t)A^{n-1}$$ where the coefficients $\alpha_i$ could ...
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Writing matrices as power series

I was reading a proof of the Cayley-Hamilton theorem here, in which the author used matrices in power series. I found out that one can write $$\left( \frac{1}{m} I + A \right)^{-1}$$ as a power ...
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Dimension of $M_{n \times n}$ over $\mathbb{R}$

Let $M_{n \times n}$ be the set of all $n$-square matrices and the characteristics polynomial of each $A\in M_{n \times n}$ is of the form $a_nt^n+a_{n-1}t^{n-1}+...+a_1t+a_0$. Then the dimension of $...
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10answers
235 views

Prove that $A^n = nA - (n-1)I$

Let $$ A = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$$ Prove that $$A^n = nA - (n-1)I$$ where $I$ is identity matrix. I have tried solving it using the Cayley-Hamilton theorem, getting $$...
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362 views

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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1answer
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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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1answer
105 views

Exponential of a Jordan Block using Cayley-Hamilton Theorem

For the sake of simplicity, I will only consider $2\times2$ matrices. The Cayley-Hamilton theorem allows us to conclude that $$e^{At} = \alpha_0I + \alpha_1 A$$ where $\alpha_0$ and $\alpha_1$ can ...
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3answers
218 views

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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1answer
127 views

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 ...
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1answer
191 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
120 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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2answers
124 views

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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2answers
138 views

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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1answer
52 views

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
205 views

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 ...
3
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1answer
168 views

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 ...