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Just watch this space!!!! :D

If it continues it will be a first! :o

khunlungphudhu, I love your avatar, any significance?

(ps in a vain hope to re-start the thread! is this CPR for a thread?)

There's only two choices really when you think about it. Either CPR or 'The Kiss of Death'. :D

Yours has turned out to be CPR but this post of mine will probably be the latter. :D

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At the far right are several large areas of activity in numerical linear algebra and related topics, typically, the study of individual matrices or transformations between (large-dimensional) real vector spaces. Numerical linear algebra per se (e.g. the determination of fast methods of solving thousands of simultaneous linear equations, the stability of eigenvalue calculations, applications to finite-element methods, sparse matrix techniques) are parts of 65: Numerical Analysis, particularly 65F: Numerical linear algebra. Here we include several fields of inquiry into the underlying linear systems and their applications. In this category we might include 15A06: Linear equations, 15A09: Matrix inversion and generalized inverses, 15A18: Eigenvalues and singular values, 15A23: Factorization of matrices (SVD, LU, QR, etc.), 15A12: Conditioning of matrices, as well as applications to physics (15A90), Control Theory (93) and Statistics (62) such as what is there known as Principal Component Analysis. Related topics include those of importance in 90: Operations Research (especially linear programming) such as 15A48: Positive matrices, 15A39: Linear inequalities, and 15A45: Miscellaneous matrix inequalities.

The circles in shades of red in the lower part of the graph show connections with other fields of algebra. Furthest down is 15A72: Invariant theory and tensor algebra, which crosses to the study of invariants in Group Theory (20) and in polynomial rings (13: Commutative Algebra and 14: Algebraic Geometry). Certain sets of matrices form well-known groups, particularly the Lie groups (22) and algebraic groups. Closer to the center of the picture are connections with Number Theory (10 and 11), especially 15A63: Quadratic forms. There are several connections with ring theory (16: Noncommutative Rings, 17: Nonassociative Rings, 19: Algebraic K-Theory); indeed many of the key examples of such rings involve collections of matrices, including the full matrix rings and Lie rings, and rings of matrices are used for representing groups and general rings. Related disciplines within Linear Algebra include 15A27: Commutativity, 15A30: Algebraic systems of matrices, 15A33: Matrices over special rings (including 12: Fields), 15A36: Matrices of integers, 15A75: Grassmann algebras, and 15A78: Other algebras. Tensor products in linear algebra (15A69) mirror such constructs in other algebraic categories.

Nearby are several fields in discrete mathematics, including the use of matrices for the representation of combinatorical objects such as graphs (05: Combinatorics), extremal matrices, permanents (15A15), and applications to 68:Computer Science, 94: Information Theory (e.g. linear codes), and 39: Difference equations.

In the upper left are the papers in "geometric algebra", including 15A66 (Clifford algebras), 81 (Quantum theory), 53 (Differential Geometry), and 58 (Analysis on manifolds).

In the upper right are the topics appropriate for 60: Probability and 62: Statistics, including 15A51: Stochastic matrices and 15A52: Random matrices, and applications to statistical mechanics (82) and the sciences (92).

Linear maps of geometric interest are considered in the geometry pages (51, 52). For example, rotation matrices and affine changes of coordinates come under 51F15. Sets of matrices qua sets arise geometrically as well; for example certain families of matrices form manifolds, and even topological groups (22).

Subfields

There is only one division (15A) but it is subdivided:

15A03: Vector spaces, linear dependence, rank

15A04: Linear transformations, semilinear transformations

15A06: Linear equations

15A09: Matrix inversion, generalized inverses

15A12: Conditioning of matrices, See also 65F35

15A15: Determinants, permanents, other special matrix functions, See also 19B10, 19B14

15A18: Eigenvalues, singular values, and eigenvectors

15A21: Canonical forms, reductions, classification

15A22: Matrix pencils, See also 47A56

15A23: Factorization of matrices

15A24: Matrix equations and identities

15A27: Commutativity

15A29: Inverse problems [new in 2000]

15A30: Algebraic systems of matrices, See also 16S50, 20Gxx, 20Hxx

15A33: Matrices over special rings (quaternions, finite fields, etc.)

15A36: Matrices of integers, See also 11C20

15A39: Linear inequalities

15A42: Inequalities involving eigenvalues and eigenvectors

15A45: Miscellaneous inequalities involving matrices

15A48: Positive matrices and their generalizations; cones of matrices

15A51: Stochastic matrices

15A52: Random matrices

15A54: Matrices over function rings in one or more variables

15A57: Other types of matrices (Hermitian, skew-Hermitian, etc.)

15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory, See also 65F35, 65J05

15A63: Quadratic and bilinear forms, inner products See mainly 11Exx

15A66: Clifford algebras, spinors

15A69: Multilinear algebra, tensor products

15A72: Vector and tensor algebra, theory of invariants, See Also 13A50, 14D25

15A75: Exterior algebra, Grassmann algebras

15A78: Other algebras built from modules

15A90: Applications of matrix theory to physics

15A99: Miscellaneous topics

Quite an interesting summary. Could you, by chance, post the graphic diagrams the text is describing?

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At the far right are several large areas of activity in numerical linear algebra and related topics, typically, the study of individual matrices or transformations between (large-dimensional) real vector spaces. Numerical linear algebra per se (e.g. the determination of fast methods of solving thousands of simultaneous linear equations, the stability of eigenvalue calculations, applications to finite-element methods, sparse matrix techniques) are parts of 65: Numerical Analysis, particularly 65F: Numerical linear algebra. Here we include several fields of inquiry into the underlying linear systems and their applications. In this category we might include 15A06: Linear equations, 15A09: Matrix inversion and generalized inverses, 15A18: Eigenvalues and singular values, 15A23: Factorization of matrices (SVD, LU, QR, etc.), 15A12: Conditioning of matrices, as well as applications to physics (15A90), Control Theory (93) and Statistics (62) such as what is there known as Principal Component Analysis. Related topics include those of importance in 90: Operations Research (especially linear programming) such as 15A48: Positive matrices, 15A39: Linear inequalities, and 15A45: Miscellaneous matrix inequalities.

The circles in shades of red in the lower part of the graph show connections with other fields of algebra. Furthest down is 15A72: Invariant theory and tensor algebra, which crosses to the study of invariants in Group Theory (20) and in polynomial rings (13: Commutative Algebra and 14: Algebraic Geometry). Certain sets of matrices form well-known groups, particularly the Lie groups (22) and algebraic groups. Closer to the center of the picture are connections with Number Theory (10 and 11), especially 15A63: Quadratic forms. There are several connections with ring theory (16: Noncommutative Rings, 17: Nonassociative Rings, 19: Algebraic K-Theory); indeed many of the key examples of such rings involve collections of matrices, including the full matrix rings and Lie rings, and rings of matrices are used for representing groups and general rings. Related disciplines within Linear Algebra include 15A27: Commutativity, 15A30: Algebraic systems of matrices, 15A33: Matrices over special rings (including 12: Fields), 15A36: Matrices of integers, 15A75: Grassmann algebras, and 15A78: Other algebras. Tensor products in linear algebra (15A69) mirror such constructs in other algebraic categories.

Nearby are several fields in discrete mathematics, including the use of matrices for the representation of combinatorical objects such as graphs (05: Combinatorics), extremal matrices, permanents (15A15), and applications to 68:Computer Science, 94: Information Theory (e.g. linear codes), and 39: Difference equations.

In the upper left are the papers in "geometric algebra", including 15A66 (Clifford algebras), 81 (Quantum theory), 53 (Differential Geometry), and 58 (Analysis on manifolds).

In the upper right are the topics appropriate for 60: Probability and 62: Statistics, including 15A51: Stochastic matrices and 15A52: Random matrices, and applications to statistical mechanics (82) and the sciences (92).

Linear maps of geometric interest are considered in the geometry pages (51, 52). For example, rotation matrices and affine changes of coordinates come under 51F15. Sets of matrices qua sets arise geometrically as well; for example certain families of matrices form manifolds, and even topological groups (22).

Subfields

There is only one division (15A) but it is subdivided:

15A03: Vector spaces, linear dependence, rank

15A04: Linear transformations, semilinear transformations

15A06: Linear equations

15A09: Matrix inversion, generalized inverses

15A12: Conditioning of matrices, See also 65F35

15A15: Determinants, permanents, other special matrix functions, See also 19B10, 19B14

15A18: Eigenvalues, singular values, and eigenvectors

15A21: Canonical forms, reductions, classification

15A22: Matrix pencils, See also 47A56

15A23: Factorization of matrices

15A24: Matrix equations and identities

15A27: Commutativity

15A29: Inverse problems [new in 2000]

15A30: Algebraic systems of matrices, See also 16S50, 20Gxx, 20Hxx

15A33: Matrices over special rings (quaternions, finite fields, etc.)

15A36: Matrices of integers, See also 11C20

15A39: Linear inequalities

15A42: Inequalities involving eigenvalues and eigenvectors

15A45: Miscellaneous inequalities involving matrices

15A48: Positive matrices and their generalizations; cones of matrices

15A51: Stochastic matrices

15A52: Random matrices

15A54: Matrices over function rings in one or more variables

15A57: Other types of matrices (Hermitian, skew-Hermitian, etc.)

15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory, See also 65F35, 65J05

15A63: Quadratic and bilinear forms, inner products See mainly 11Exx

15A66: Clifford algebras, spinors

15A69: Multilinear algebra, tensor products

15A72: Vector and tensor algebra, theory of invariants, See Also 13A50, 14D25

15A75: Exterior algebra, Grassmann algebras

15A78: Other algebras built from modules

15A90: Applications of matrix theory to physics

15A99: Miscellaneous topics

Quite an interesting summary. Could you, by chance, post the graphic diagrams the text is describing?

And the source of the info :o

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And the source of the info

so , i see you are an algebra fan also , good for you , a splendid way to pass the time.

the source for my post is here ,

http://www.math.niu.edu/~rusin/known-math/index/15-XX.html

where you will find the diagram referred to.

if you found that interesting ,although personally i found it a bit basic , you might want to check out some of the following

Independence of the vector space axioms

History of the Cauchy-Schwarz inequality

The mysteries of taxexillian transmutation matrices

Direct rotation -- shortest rotation taking one vector to another

Proof of the Cayley-Hamilton Theorem

Determinants, permanents, and immanents of a matrix

Determinant-preserving endomorphisms of End(V)

Using Dodgson's condensation formula to find determinants of symmetric Toeplitz matrices

Inverses, determinants of Vandermonde-like special matrices (and the 'Advanced determinant calculus')

Famous conjectures: existence of Hadamard matrices, projective planes, Jacobian conjecture

What is the probability that two matrices will commute

What are eigenvalues and linear transformations?

How to compute eigenvectors (after the eigenvalues) for a 3x3 matrix.

The determinant equals the product of the eigenvalues.

Eigenvalues of a symmetric matrix and the symmetric part of a general square matrix.

Multiple characterizations of positive (semi)definite matrices, and applications

Why are eigenvalues of Hermitian matrices real?

Gershgorian circles (for matrix eigenvalues)

Eigenvalues of a circulant matrix.

Generalized eigenvalue problem -- complete set of eigenvectors?

Smallest matrix norms match largest eigenvalue

The Spectral Radius Formula (for operator norm of matrices)

Distribution of eigenvalues in random matrices

Power method: successive Rayleigh quotients converge to dominant eigenvalue of a matrix

Do the parts of the Jordan Decomposition of a matrix vary continuously with the matrix? (Not really)

Is similarity achieved over the ground field? (yes)

Example of companion matrices (to Chebyshev polynomials)

Finding a symmetric matrix with prescribed characteristic polynomial

When is a matrix irreducible (no invariant subspaces)?

Generalized inverses of a matrix: definitions and applications

Led zeppelin chord progressions for the hard of hearing

The Moore-Penrose pseudo-inverse of a matrix.

Computing the pseudoinverse using SVD.

Using the pseudo-inverse and Tihonov regularization to solve linear systems of equations.

Computing determinants of Toeplitz matrices

Comparison of various factorizations of symmetric, positive definite matrices

Some pointers on the computation of the Singular Value Decomposition of a matrix.

Karhunen-Loève procedure: picks out the dominant terms of the Singular Value Decomposition (Proper Orthogonal Decomposition, Principal Component Analysis, analysis by Empirical Eigenfunctions)

Taxexiles polynomial theory of why so many ######s live on koh samui

Lay person's description of Principal Component Analysis

What is the Singular Value Decomposition

Using the Cholesky factorization of a matrix to find an isometric embedding of a finite set of points.

Not all symmetric matrices have a (modified) Cholesky-factorization

Real Polar Decomposition of a real matrix as (symmetric positive semidefinite)*(orthogonal)

Maple code to do QR decomposition of a matrix.

Bunch-Parlett matrix decomposition (and counting negative eigenvalues)

Every square real matrix is the product of two symmetric matrices

Basic code for Gauss-Jordan inversion of a square matrix.

Good algorithm for computing minimal polynomial?

Matrix inversion by Monte-Carlo techniques(!): citations.

Pointer to text on Matrix Algorithms [G B "Pete" Stewart]

Efficient (recursive) methods of matrix multiplication (Strassen algorithm)

Hints for solving a large linear system of equations.

Solving a large sparse system of linear equations.

Pointers to sparsity plots and other matrix software

What are the multiplicative scalar functions on matrices? (determinants...)

Counting annihilating matrices over a finite field.

Use of permutation groups to determine a method for transposing nonsquare matrices in place.

Counting the dimensions of magic squares and cubes.

Example of expressing a vector as a linear combination of two others.

Writing a matrix as a linear combination of orthogonal matrices.

Finding linear combinations of matrices to have rank 1

How to tell if a family of polynomials is linearly independent over Q.

Given many vectors in a vector space, how to find linear relations among few of them?

Definition and properties of the square-root function on (positive-definite) matrices

General solution of matrix quadratic equations

Lyapunov matrix equation, Sylvester matrix equation

Origin and scope of the instability associated with the Hilbert matrix

What is so ill-conditioned about the Hilbert matrix?

Computing the inverse of the ill-conditioned Hilbert matrices

Computing the determinant of the Hilbert matrices.

Kantorovich's Inequality for positive definite Hermitian matrices.

Matrix inequalities for positive definite matrices (Hadamard, Szász, etc.)

Pointers to results on non-negative matrices (Perron-Frobenius theory)

Extensions of Perron-Frobenius theorem to nonnegative matrices

Vector spaces with periodic automorphism groups

What is the tensor product of vector spaces?

Application of exterior algebras to generalize the factorization of adj(X) when X is singular

Current research trends in multilinear algebra

Some references for multilinear algebra

Application of Clifford algebras to quadratic forms

Exploring Clifford algebras and Geometric Algebra and applications.

Applications of Clifford algebras to differential topology and physics

but be warned , once you start , you wont be able to stop , its addictive.

enjoy , :o

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Thanks for the link, taxexile. However, after a closer look the author is a bit disappointing. Although he did mention 'Quantum Theory" in his summary, he obviously is just another of those selfcontained mathematical theoricians, who doesn't have the slightest clue to how his field is applied in the real world. How about Gilbert Space and its influence on Einstein's equation on Generel Relativity? How about Lie Algebra and its influence on the Hermite Function, which is very essential in the field of modern Quantum Field Theory - which is the field that is closest to understand Big Bang (the birth of this universe) as well as Big Crunch - the death of this universe - and thereby the end of not only this thread, but also the very ThaiVisa website itself?

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1. Off topic remarks.

2. Personal attacks.

3. Criticism of a poster's spelling, grammar or girlfriend.

4. Comments about gut-bucket farangs.

5. Comments about old farangs.

6. Comments about bg's.

7. Questions about the best way to own land in LOS even though the law says you can't.

I assume that you have a gut-bucket and are old :o only joking.

Does anyone know if a Thai lady can actually buy land after she marries a farang?

I think she keeps all her rights, although many disagree. I believe that if she were a BG, then she will get a hard time, but if she has connections she can.

By the way does anyone know who Chelsea are playing tonight?

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I dispute vigourously your implied assumption that you are TV's top thread killer. I aspire to that noble position myself. When I say something, the thread stops, I get totally ignored or I get booted.

God I love TV

If it means that much to you, I'll answer as well. How are you OlRedEyes?

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1. Off topic remarks.

2. Personal attacks.

3. Criticism of a poster's spelling, grammar or girlfriend.

4. Comments about gut-bucket farangs.

5. Comments about old farangs.

6. Comments about bg's.

7. Questions about the best way to own land in LOS even though the law says you can't.

I assume that you have a gut-bucket and are old :o only joking.

Does anyone know if a Thai lady can actually buy land after she marries a farang?

I think she keeps all her rights, although many disagree. I believe that if she were a BG, then she will get a hard time, but if she has connections she can.

By the way does anyone know who Chelsea are playing tonight?

A Thai lady can buy land after she marries a farang. I know one who has done it! The farang husband must attend the purchase formalities at the Land Office and sign a statement to the effect that the land and the money that's buying it are not his and that he has nothing to do with it.

I like to think that I'm not a gut bucket and I'm as old as the woman that I ....... I'm actually perfectly formed in every detail and one of those lucky people who just never age! And I always tell the truth :D

Chelsea? I haven't heard from her in ages!

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daleyboy

in his summary, he obviously is just another of those selfcontained mathematical theoricians, who doesn't have the slightest clue to how his field is applied in the real world.

i know these mathematical papers can take some ploughing through but i dont think you could have read it thoroughly my friend , if led zepp aint the real world then what is. :o

Is similarity achieved over the ground field? (yes)

Example of companion matrices (to Chebyshev polynomials)

Finding a symmetric matrix with prescribed characteristic polynomial

When is a matrix irreducible (no invariant subspaces)?

Generalized inverses of a matrix: definitions and applications

Led zeppelin chord progressions for the hard of hearing

The Moore-Penrose pseudo-inverse of a matrix.

Computing the pseudoinverse using SVD.

Using the pseudo-inverse and Tihonov regularization to solve linear systems of equations.

Computing determinants of Toeplitz matrices

Comparison of various factorizations of symmetric, positive definite matrices

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I dispute vigourously your implied assumption that you are TV's top thread killer. I aspire to that noble position myself. When I say something, the thread stops, I get totally ignored or I get booted.

God I love TV

If it means that much to you, I'll answer as well. How are you OlRedEyes?

I am very well thank you, Suegha. Thanks for asking.

Though my haemeroids do hurt a little these days....

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I like to think that I'm not a gut bucket and I'm as old as the woman that I ....... I'm actually perfectly formed in every detail and one of those lucky people who just never age! And I always tell the truth :o

Chelsea? I haven't heard from her in ages!

Do describe the perfect form for us..... :D

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A Thai lady can buy land after she marries a farang. I know one who has done it! The farang husband must attend the purchase formalities at the Land Office and sign a statement to the effect that the land and the money that's buying it are not his and that he has nothing to do with it.

I too know a girl who just bough many rai, but her father is an MP and ex policeman.

I heard from a friend that his wife was refused recently.

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A Thai lady can buy land after she marries a farang. I know one who has done it! The farang husband must attend the purchase formalities at the Land Office and sign a statement to the effect that the land and the money that's buying it are not his and that he has nothing to do with it.

I too know a girl who just bough many rai, but her father is an MP and ex policeman.

I heard from a friend that his wife was refused recently.

I saw your post on the other thread. My wife bought land last year and I signed the usual declaration. There was no help or influence from elsewhere. I think that there must be more to the story that you heard.

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I'm pretty good at killing threads. But I've got a lot of advantages over most.

1. I'm boring.

2. I have tons of experience talking to myself.

3. I'm long tedious, rhapsodic, periphrastic, declamatory, bombastic, diffuse, repetitious, fustian, balderdash, gabby, involved, tautological, long-winded, magniloquent, prolix, rhetorical, talky, tortuous, windy, yacking, aureate, big-talking, fustian, grandiloquent, grandiose, high-flown, highfalutin, histrionic, inflated, magniloquent, tumid, orotund, ranting, swollen, turgid, verbose, windbag, windy, repeating, wordy, circumlocutory, flowery, tautologous, repetitive, pleonastic, talkative, bullshitting, euphuistic, ostentatious, loudmouth, flowery, palaverous, garrulous, redundant, sonorous, overblown, grandiloquent, loquacious, diffusive, and I ramble a lot, too.

4. I use lots of big words so people can't understand what I'm talking about.

5. I have the personality of a wombat.

6. Nobody likes me so if I post too much people start avoiding me like the plague and everyone eventually drops out of the thread. It gets really bad when I have the last two posts in a row.

7. I never have anything nice to say about anyone.

8. I usually wait until the thread is at least a few weeks old.

Is anyone reading this? :D:o

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I'm pretty good at killing threads. But I've got a lot of advantages over most.

1. I'm boring.

2. I have tons of experience talking to myself.

3. I'm long tedious, rhapsodic, periphrastic, declamatory, bombastic, diffuse, repetitious, fustian, balderdash, gabby, involved, tautological, long-winded, magniloquent, prolix, rhetorical, talky, tortuous, windy, yacking, aureate, big-talking, fustian, grandiloquent, grandiose, high-flown, highfalutin, histrionic, inflated, magniloquent, tumid, orotund, ranting, swollen, turgid, verbose, windbag, windy, repeating, wordy, circumlocutory, flowery, tautologous, repetitive, pleonastic, talkative, bullshitting, euphuistic, ostentatious, loudmouth, flowery, palaverous, garrulous, redundant, sonorous, overblown, grandiloquent, loquacious, diffusive, and I ramble a lot, too.

4. I use lots of big words so people can't understand what I'm talking about.

5. I have the personality of a wombat.

6. Nobody likes me so if I post too much people start avoiding me like the plague and everyone eventually drops out of the thread. It gets really bad when I have the last two posts in a row.

7. I never have anything nice to say about anyone.

8. I usually wait until the thread is at least a few weeks old.

Is anyone reading this? :D:o

You missed one:

9. I'm articulate.

That usually works well. :D

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