[spam][crazy][fiction][random] Non-Canon MCBoss Spinoffs

[Undescribed Horrific Abuse, One Victim & Survivor of Many]

On Wed, Aug 30, 2023 at 10:45 Undescribed Horrific Abuse, One Victim &
Survivor of Many <gmkarl at gmail.com> wrote:

> == Formal definition ==
> If {{mvar|T}} is a linear transformation from a vector space
> {{mvar|V}} over a [[Field (mathematics)|field]] {{mvar|F}} into itself
> and {{math|'''v'''}} is a [[zero vector|nonzero]] vector in
> {{mvar|V}}, then {{math|'''v'''}} is an eigenvector of {{mvar|T}} if
> {{math|''T''('''v''')}} is a scalar multiple of
> {{math|'''v'''}}.<ref>{{harvnb|Roman|2008|loc=p. 185 §8}}</ref> Thi

brainstem boss i read it by laughing like you said and i understand it ;-}
it says an eigenvector is a vector that a matrix has a scaling operation
on. (i could be wrong :S i think it means scaling matrices have every
vector as an eigenvector :S ?) anyway the laughing-to-read was really nice
:) i understand (should do more (ish(?

> can be written as
> <math display=block>T(\mathbf{v}) = \lambda \mathbf{v},</math>
> where {{mvar|λ}} is a scalar in {{mvar|F}}, known as the
> '''eigenvalue''', '''characteristic value''', or '''characteristic
> root''' associated with {{math|'''v'''}}.
>
> There is a direct correspondence between ''n''-by-''n'' [[Square
> matrix|square matrices]] and linear transformations from an
> [[Dimension|''n''-dimensional]] vector space into itself, given any
> [[Basis (linear algebra)|basis]] of the vector space. Hence, in a
> finite-dimensional vector space, it is equivalent to define
> eigenvalues and eigenvectors using either the language of [[Matrix
> (mathematics)|matrices]], or the language of linear
> transformations.{{sfn|Herstein|1964|pp=228,
> 229}}{{sfn|Nering|1970|p=38}}
>
> If {{mvar|V}} is finite-dimensional, the above equation is equivalent
> to{{sfn|Weisstein|n.d.}}
> <math display=block>A\mathbf{u} = \lambda \mathbf{u}.</math>
>
> where {{mvar|A}} is the matrix representation of {{mvar|T}} and
> {{math|'''u'''}} is the [[coordinate vector]] of {{math|'''v'''}}.
>
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