2
votes
2answers
86 views

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 $A$ ...
4
votes
3answers
67 views

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 $\mathbf{x}$ ...
1
vote
2answers
127 views

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 ...
1
vote
1answer
41 views

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 ...
6
votes
3answers
2k views

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 ...
8
votes
1answer
1k views

intuition for complex eigenvalues

The eigenvalues of a rotation matrix are complex numbers. I understand that they cannot be real numbers because when you rotate something no direction stays the same. My question What is the ...
9
votes
6answers
4k views

How are eigenvectors/eigenvalues and differential equations connected?

In school and at university we never had eigenvalues nor differential equations, so these concepts were really giving me a hard time. Now I developed some intuition for both concepts. I learned that ...