Conservapedia - User contributions [en]

Eigenvalue

2010-05-02T14:58:40Z

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{{merge|eigenvector}}
An '''eigenvalue''' of a square [[matrix]] <math>A</math> is a [[real|real number]] or [[complex number]] <math>\lambda</math> such that

:<math>A\boldsymbol{x}=\lambda\boldsymbol{x}</math>

for some non-zero vector <math>\boldsymbol{x}\in\mathbb{R}</math> known as a [[eigenvector]]. The eigenvalues are the zeroes of a matrix's [[characteristic polynomial]].

The product of all the eigenvalues of a matrix equals the value of the matrix's determinant. Since a matrix is invertible if and only if the determinant is non-zero, it is invertible if and only if zero is not an eigenvalue.

The [[span]] of all the eigenvectors corresponding to a fixed eigenvalue <math>\lambda</math> is called the [[eigenspace]] <math>E_\lambda</math> of <math>A</math>.

[[category:Linear algebra]]

Cary Grant

2010-05-02T14:27:10Z

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'''Cary Grant''' (born Archibald Alexander Leach; 18 January, 1904 – 29 November, 1986) was a 20th century movie actor well known for his romantic and comedic roles, although justly famous for his star turn in [[Alfred Hitchcock's]] "[[North By Northwest]]."

Born in [[Bristol]], [[England]], he joined a stage troupe after being expelled from school in 1918. He went with them to the [[United States]] in 1920 and remained there, working as a lifeguard on Coney Island. In 1923, he returned to England anf played small parts in musical comedies. After a few years, he returned to America. He played parts in several films in 1932, notably ''[[Blonde Venus]]'' with [[Marlene Dietrich]].

His first starring role, in 1933, was in ''[[She Done Him Wrong]]'' with [[Mae West]]; she famously said to him "Why don't you come up sometime ‘n see me?". After that, he was established as a straight romantic leading man. In 1937-8 he demonstrated his flair for screwball comedy in ''[[Topper]]'', ''[[Bringing Up Baby]]'' and [[Holiday (film)|Holiday]]''. Always seeking to diversify, he played several dramatic roles for Hitchcock. He retired from films in 1966. He never received an Oscar (although nominated twice), but was given an honorary one in 1970.

In "[[People Will Talk]]" (1951), he plays an outspoken but kind medical professor.

[[category:Actors]]

Cary Grant

2010-05-02T12:56:18Z

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Cary Grant

2010-05-01T17:37:40Z

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Companion bill or measure

2010-05-01T17:32:04Z

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This is a term in [[United States]] legislative practice. It refers to similar or identical legislation which is introduced in the [[United States Senate|Senate]] and [[United States House of Representatives|House]]. House and Senate lawmakers who share similar views on legislation may introduce a companion [[bill]] in their respective chambers to promote simultaneous consideration of the [[measure]].<ref>[http://www.senate.gov/reference/glossary_term/companion_bill.htm] US Senate Reference</ref>

==References==
<references/>

[[Category:United States Senate Terms]]

Hesperides

2010-05-01T17:27:19Z

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The '''Hesperides''' (or "African Sisters') of [[Greek mythology]] are [[nymph]]s who live in a beautiful garden, said to be situated either in the Arcadian Mountains of Greece or, near the western extremes of the [[Mediterranean]], near [[Mount Atlas]].

In this garden grows the tree with the golden apples, which [[Gaia]] had given as a wedding present to [[Hera]], after she married [[Zeus]]. This garden and the tree are guarded by [[Ladon]], a dragon with a hundred heads. as the eleventh of his twelve labours, [[Hercules]] managed to steal some of the apples, by tricking [[Atlas]] into picking them for him.

The Hesperides are named as Aegle, Arethusa, Erytheia and Hesperia. <ref>http://www.pantheon.org/articles/h/hesperides.html</ref>

==References==
* Graves, Robert (1955): The Greek Myths: Pub. Pelican
{{Reflist}}
{{Nb mythology greece}}
[[category:Greek Mythology]]

Characteristic polynomial

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The '''characteristic polynomial''' of a square [[matrix]] <math>A</math> is given by,

:<math>f(\lambda) = \det(A-\lambda I)</math>

where <math>I</math> is the [[identity matrix]]. The [[root]]s of this [[polynomial]] are the [[eigenvalue]]s of the matrix.

The degree of this polynomial equals the length of the matrix's side. The number of roots therefore also equals this number, though some roots may be repeated.

[[Category:Linear algebra]]

Eigenvalue

2010-05-01T17:18:41Z

Versatile: Minor corrections

{{merge|eigenvector}}
An '''eigenvalue''' of a square [[matrix]] <math>A</math> is a [[real|real number]] or [[complex number]] <math>\lambda</math> such that

:<math>A\boldsymbol{x}=\lambda\boldsymbol{x}</math>

for some non-zero vector <math>\boldsymbol{x}\in\mathbb{R}</math> known as a [[eigenvector]]. The eigenvalue is the zero of a matrix's [[characteristic polynomial]].

The [[span]] of all the eigenvectors corresponding to a fixed eigenvalue <math>\lambda</math> is called the [[eigenspace]] <math>E_\lambda</math> of <math>A</math>.

[[category:Linear algebra]]

Talk:Bible: Textual Criticism

2010-05-01T17:16:43Z

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Because this subject can involve some heated debate, some rules have to be set down, to wit:

*The Bible is to be respected, and by extention, the beliefs of Jews and Christians regarding it.

*The people writing here will be respected, and it doesn't matter which side of the fence they're on in the subject in question. If you disagree with a writer, be respectful and explain why.

*The article is to be well written with valid source material; the purpose of the article is to explain both sides of textual criticism, and it will be the best article written on the subject...nothing less will do. [[User:Karajou|Karajou]] 12:28, 20 March 2007 (EDT)

I've added three very rerspectable sources and can add more if it's helpful.--[[User:Versatile|Versatile]] 13:16, 1 May 2010 (EDT)

Bible: Textual Criticism

2010-05-01T17:14:37Z

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This page is for the future article about the historical textual criticism of the [[Bible]] (see talk page)

== Bibliography ==
* Bruce, F. F. (1991): The Books and the Parchments (5th ed0: Marshall Pickering
* Metzger, Bruce (1992): The Text of the New Testament (3rd ed): Oxford University Press
* Roberts, Bleddyn (1951): The Old Testament Text and Versions: University of Wales Press

[[Category:Bible]]

Linear algebra

2010-05-01T17:03:59Z

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'''Linear algebra''' is the branch of [[mathematics]] that deals with [[vector]]s, vector spaces, linear maps, and systems of linear [[equation]]s. Topics studied in Linear algebra include Linear equations, Matrices, [[Matrix]] decompositions, Computations, Vectors, Vector spaces, Multilinear algebra, Affine space, Projective space.<ref>[http://www.m-w.com/cgi-bin/dictionary?book=Dictionary&va=linear+algebra Merriam Webster definition of Linear Algebra]</ref>
Linear algebra has numerous applications in [[engineering]], [[chemistry]] and [[physics]].

The three most vexing computations in linear algebra are these:
#linear equations
#least squares
#finding the eigenvalues of an ''n x n'' matrix for n > 3.

== Basic Concepts ==

*[[Linear equations]]
:*coefficient matrices and Gauss-Jordan elimination
:*[[rank]] (row rank = column rank)
:*geometric representations, especially [[vector]]s and systems with more variables than equations

*[[Linear transformations]]
:*transformations, inverses and matrix products

*Subspaces
:*[[image]] and [[kernel]]
:*[[basis]] and [[span]]
:*[[dimension]]

*[[Determinant]]s
:*[[Sarrus's rule]]
:*geometrical interpretation
::*classical adjoint, expansion factor, application to [[parallelepiped]]s)
:*determinant of similar matrix, [[inverse matrix]], product of matrices
:*Cramer's rule (with and without product rule)
:*[[minor]] of a matrix
:*Laplace expansion (cofactors)

*[[Basis]]
:*[[eigenbasis]]
:*[[orthogonal]] eigenbasis
:*[[orthonormal]] eigenbasis

*[[Eigenvalue]]s and [[eigenvector]]s
:*diagonalization
:*[[trace]]
:*[[characteristic polynomial]]
:*complex eigenvalues

*[[Orthogonality]]
:*orthogonal (perpendicular) vectors
:*orthonormal vectors
:*orthogonal projections
:*[[orthogonal matrix]]
:*projections
:*Gram-Schmidt Process and QR Factorization
:*orthogonal matrices, orthogonal transformations
:*data fitting, especially least squares

*[[Symmetric matrix]]
:*[[spectral theorem]]

*Quadratic forms

*Linear dynamical systems
:*Euler's Formula

== More advanced topics include ==

*[[Vector space]]s (linear spaces)
:*the conditions of a vector space
:*isomorphisms
:*Nth dimensional spaces and subspaces

*Inner spaces
:*inner product spaces

*Determinants
:*cofactor
:*adjugate (useful in finding the inverse of a matrix)

*[[Cayley-Hamilton theorem]]

*Stability

*Hermitian Matrices

*Singular values and [[Singular Value Decomposition]]

*Linear differential equations

== Common problems ==

Common problems in linear algebra include:

*simplifying or reducing matrices
*[[Gauss-Jordan elimination]]
*matrix multiplication
*finding inverses and transposes of matrices
*[[Gram-Schmidt procedure]]
*finding [[eigenvalue]]s and [[eigenvector]]s for matrices:
:#find the [[characteristic polynomial]]
:#solve for the eigenvalues
:#solve for the eigenvectors
*finding the orthogonal projection of a vector in a vector space
*diagonalize a matrix
*find the geometric equivalent of a matrix
*finding the determinant of a 2x2 matrix (easy) and a 3x3 matrix (hard)
*finding the inverse of a matrix
*[[decomposition]] or [[factorization]] of a matrix: representing a given matrix as a product of simpler matrices

==References==
{{reflist}}

[[Category:Linear algebra]]
[[category:mathematics]]