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