# Tagged Questions

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### What's the fastest way to determine Eigenvalues & Eigenvectors of any 2 by 2 Matrix?

My instructor claims that it's inefficient and superfluous to compute eigenvectors de novo for each $2$ by $2$ matrix. He suggested a trick instead which resembles the eigenvectors and cases here. ...
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### Eigevectors of an Idempotent Matrix: A^2 = A [Strang P310 6.2.25]

Source: P4 of http://www.minho-kim.com/courses/11sp71007/data/hw05-solution.pdf $\Large{{1.}}$ I perceive : Because $A\mathbf{a_i = 1a_i}$ for all $1 \le i \le n$, thus $1$ is an eigenvalue of $A$ ...
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### Any vector is a linear combination of the eigenvectors ? [Strang P296 6.1.25]

Suppose $A$ and $B$ have the same eigenvalues $\lambda_1, \cdots, \lambda_n$ with the same independent eigenvectors $\mathbf{x_1, \cdots, x_n}$. Then $A = B$. Reason: Any vector $\mathbf{x}$ ...
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### Confused with Eigenvalues and Eigenvectors and Vector transformations

Hello fellow mathematicians, I am studding " Eigenvalues and Eigenvectors " at this point and I need to understand something here: I am actually performing automatic operations on finding them, but ...
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### Why is $E_{\lambda}$ the kernel of the linear map $\alpha-\lambda I$

The book starts the chapter on Eigenvalues and Eigenvectors, and goes that this statement is obvious. Here $E_{\lambda}$ stands for the set of vectors $v$ such that $α(v) = λv$, for any scalar ...
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### Examples for proof of geometric vs. algebraic multiplicity

Here you see a supposedly easy proof of a well-known theorem in linear algebra: Although I know I should understand this, I don't :-( Obviously there are too many indices and stuff, so I don't see ...