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1. ### I Eigen Vectors, Geometric Multiplicities and more....

=> A is diagonalizable : ##A \sim \begin{pmatrix}\lambda_1 \text{ Id}_{m_1} & 0 & 0 \\ 0 & \ddots & 0\\ 0 & 0 & \lambda_p \text{ Id}_{m_p} \end{pmatrix}##. What is ##m_1,...,m_p## ? What is ##m_1 + ... + m_p ## equal to ? <= Say that matrix A represents an endomorphism on vector space ##E##...
2. ### Cofactors & Determinants

Look at things this way : given an ##n\times n## matrix ##A##, with real coefficients for exemple, its determinant is the determinant of the column vectors in the canonical basis ##{\cal B}## of ##M_{n,1}(\mathbb{R})##, which you could write ## \det A = \det_{\cal B} (C_1,...,C_n)##. With the...
3. ### Insights What Are Eigenvectors and Eigenvalues? - Comments

Nice insight ! If you like it, I have an exemple of application for your post to euclidean geometry. You could explain how eigenvalues and eigenvectors are helpfull in order to carry out a full description of isometries in dimension 3, and conclude that they are rotations, reflections, and the...
4. ### Reduction of endomorphisms

Finally! That made me sweat ;-) Thank you very much Samy for your patience
5. ### Reduction of endomorphisms

Think I have it now. In a previous post, we said that ##P(f) = 0## implies that the eigenvalues of ##f## are among the zeros of ##P##. Then it had to be true that ## P(f) = 0 \iff Q(f) = 0 ##, where ##Q = \prod_{\lambda \in \text{Sp}(f)} (X - \lambda) ##. Now I want to show that ## E =...
6. ### Reduction of endomorphisms

But I'm not giving up on this, I want to find the solution :-)
7. ### Reduction of endomorphisms

Lol, this is exactly the reason why I deleted my post :-) I need to work more on this. Thank you for your help

9. ### Reduction of endomorphisms

Oooops, sorry it's me who's wasting your time, I realize I am completely confused with the notations. I need to re-do this. Sorry
10. ### Reduction of endomorphisms

If ##P(f) = 0##, then ##P(f)(x) = 0 ## for all ##x\in E##. If ##x = 0## then ##x \in E \cap \bigoplus_{\lambda \in \text{Sp}(f) } E_{f,\lambda}## If ##x\neq 0##, then ## 0 = P(f)(x) = a\prod_{i = 1}^p (f(x) - \lambda_i x)##. So at least one term in the product is equal to 0. Therefore, there...
11. ### Reduction of endomorphisms

Hello, I am studying reduction of endomorphisms and I came across a theorem that I can't understand completely. It states that: Theorem: Let ##E## be a finite dimensional ##K## vector space, ##K## sub-field of ##\mathbb{C}##, and ##f## be an endomorphism of ##E##. Then, ##f## is diagonalizable...
12. ### A reasonable analogy for understanding similar matrices?

Isn't similarity an equivalence relation between square matrices ?
13. ### Abstract Algebra: Dummit and Foote Exercise

It rather follows from ##\sigma_i## being a cycle. It says that a permutation can be written as a product of disjoint cycle ##\sigma = \sigma_1 \circ ... \circ \sigma_p ##. The question asks you to find the smallest ##m \ge 1## such that ##\sigma ^ m = \text{id}##. Since the cycles are...