# 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 ...
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1 vote
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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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1 vote
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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,...
1 vote
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 ...
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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 ...
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1 vote
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 ...
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1 vote
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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 ...
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1 vote
203 views

### 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 ...
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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 ...
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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 ...
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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
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