# 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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1 vote
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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 ...
1 vote
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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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### 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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1 vote
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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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1 vote
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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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### 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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