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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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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 ...
user1274233's user avatar
1 vote
0 answers
39 views

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 ...
Kishalay Sarkar's user avatar
4 votes
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80 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 ...
Maverick's user avatar
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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. ...
Jim Newton's user avatar
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Let $A \in M_n(\mathbb R)$ such that $A^2+A+5I_n = 0$. Find the characteristic polynomial of $A$ [duplicate]

Let $A \in M_n(\mathbb R)$ such that $$A^2+A+5I_n = 0$$ Find the characteristic polynomial of $A$. I tried two different approaches and got stuck on both. I am wondering if I was even headed in the ...
MathStudent101's user avatar
0 votes
1 answer
127 views

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 ...
strugglingStudent's user avatar
0 votes
0 answers
63 views

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 ...
Dumbest person on earth's user avatar
1 vote
3 answers
1k views

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)$0$ ...
Maverick's user avatar
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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+...
Maverick's user avatar
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1 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 ...
User626's user avatar
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2 answers
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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,...
strugglingStudent's user avatar
1 vote
0 answers
109 views

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 ...
RdFg's user avatar
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5 votes
1 answer
239 views

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 ...
Stefan Solomon's user avatar
1 vote
1 answer
112 views

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 ...
Stefan Solomon's user avatar
1 vote
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 ...
Bill's user avatar
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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 ...
Stefan Solomon's user avatar
7 votes
1 answer
233 views

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 $...
Stefan Solomon's user avatar
3 votes
0 answers
114 views

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 ...
user264745's user avatar
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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 $T$ w.r.t. some basis of $V$. I know this is well-defined as it is ...
jet's user avatar
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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 ...
Linkin's user avatar
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1 vote
1 answer
190 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 ...
Stefan Solomon's user avatar
0 votes
1 answer
77 views

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 ...
Vasilis 's user avatar
2 votes
0 answers
112 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 ...
Henry T.'s user avatar
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2 votes
0 answers
103 views

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 ...
Tosca's user avatar
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2 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 ...
zifan ying's user avatar
2 votes
1 answer
146 views

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 $\...
Paul's user avatar
  • 1,364
4 votes
2 answers
78 views

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 ...
user2345678's user avatar
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3 votes
2 answers
227 views

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 ...
Jonas Linssen's user avatar
2 votes
0 answers
169 views

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 ...
Michał Kuczyński's user avatar
1 vote
2 answers
57 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 \...
Ussesjskskns's user avatar
0 votes
4 answers
146 views

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 ...
user avatar
2 votes
2 answers
165 views

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 ...
C-Web's user avatar
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0 answers
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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 $O$ ...
Rose2357's user avatar
1 vote
2 answers
99 views

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 ...
Paras Khosla's user avatar
  • 6,411
-1 votes
2 answers
52 views

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 ...
MAHESH T's user avatar
1 vote
0 answers
171 views

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 ...
Marion's user avatar
  • 2,229
2 votes
3 answers
213 views

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*}...
matteogost's user avatar
4 votes
1 answer
561 views

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 $...
tchappy ha's user avatar
  • 8,720
8 votes
0 answers
323 views

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 ...
Aitor Iribar Lopez's user avatar
1 vote
2 answers
328 views

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^...
Erik Eriksson's user avatar
0 votes
1 answer
113 views

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.
Governor's user avatar
  • 469
1 vote
1 answer
68 views

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 &...
Jake Stevens-Haas's user avatar
0 votes
0 answers
97 views

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 ...
Balazs's user avatar
  • 129
1 vote
1 answer
91 views

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 ...
94thomas's user avatar
2 votes
0 answers
263 views

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 ...
Tomer's user avatar
  • 436
1 vote
2 answers
126 views

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 ...
john's user avatar
  • 1,288
3 votes
0 answers
173 views

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, ...
FShrike's user avatar
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0 votes
4 answers
870 views

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,...
kirti purohit's user avatar
5 votes
1 answer
198 views

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 ...
Tony Stark's user avatar
2 votes
3 answers
513 views

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 ...
Bhartendu Kumar's user avatar