算盤

2011-02-09

Nodal Discontinuos Galerkin methods

http://www.caam.rice.edu/~timwar/Book/NodalDG.html
http://www.caam.rice.edu/~timwar/TimWarburton/HomePage.html

2011-02-04

dot product

Vector formulation

The law of cosines is equivalent to the formula

$\vec b\cdot \vec c = \Vert \vec b\Vert\Vert\vec c\Vert\cos \theta$

in the theory of vectors, which expresses the dot product of two vectors in terms of their respective lengths and the angle they enclose.

Fig. 10 — Vector triangle

Proof of equivalence. Referring to Figure 10, note that

$\vec a=\vec b-\vec c\,,$

and so we may calculate:

$\begin{align} \Vert\vec a\Vert^2 & = \Vert\vec b - \vec c\Vert^2 \\ & = (\vec b - \vec c)\cdot(\vec b - \vec c) \\ & = \Vert\vec b \Vert^2 + \Vert\vec c \Vert^2 - 2 \vec b\cdot\vec c. \end{align}$

The law of cosines formulated in this notation states:

$\Vert\vec a\Vert^2 = \Vert\vec b \Vert^2 + \Vert\vec c \Vert^2 - 2 \Vert \vec b\Vert\Vert\vec c\Vert\cos(\theta), \,$

which is equivalent to the above formula from the theory of vectors.

$A\cdot B=|A\|B|\cos \theta$ (by definition of dot product)

If you think of the length of the 3 vectors |A|,|B| and |B-A| as the lengths of the sides of a triangle, you can apply the law of cosines here too (To visualize this, draw the 2 vectors A and B onto a graph, now the vector from A to B will be given by B-A. The triangle formed by these 3 vectors is applied to the law of cosines for a triangle)
$C^2 = A^2 + B^2 - 2|A||B|\cos \theta$
In this case, we substitute: |B-A| for c, |A| for a, |B| for b
and we obtain:
$|B-A|^2 = |A|^2 + |B|^2 - 2 |A||B|\cos(\theta)$ (by law of cosines)

Remember now, that Theta is the angle between the 2 vectors A, B.
Notice the common term |A||B|cos(Theta) in both equations. We now equate equation (1) and (2), and obtain

$A\cdot B = (-(|B-A|^2) + |A|^2 + |B|^2) / 2$
and hence
$A\cdot B =\frac{1}{2} (-((B_x^2 - 2A_x B_x + A_x^2) + (B_y^2 - 2A_y B_y + A_y^2) + (B_z^2 - 2A_zB_z + A_z^2)) + A_x^2 + A_y^2 + A_z^2 + B_x^2 + B_y^2 + B_z^2)$
(by pythagorean length of a vector) and thus
$A\cdot B = A_xB_x + A_yB_y + A_zB_z$

cross product

http://www.physics.orst.edu/bridge/mathml/dot+cross.xhtml

2011-02-01

Rotation matrix

From Wikipedia, the free encyclopedia

In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix

$R = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{bmatrix}$

rotates points in the xy-Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system. To perform the rotation, the position of each point must be represented by a column vector v, containing the coordinates of the point. A rotated vector is obtained by using the matrix multiplication Rv (see below for details).

In two and three dimensions, rotation matrices are among the simplest algebraic descriptions of rotations, and are used extensively for computations in geometry, physics, and computer graphics. Though most applications involve rotations in two or three dimensions, rotation matrices can be defined for n-dimensional space.

Rotation matrices are always square, with real entries. Algebraically, a rotation matrix in n-dimensions is a n × n special orthogonal matrix, that is an orthogonal matrix whose determinant is 1:

$R^{T} = R^{-1}, \det R = 1\,$ .

The set of all rotation matrices forms a group, known as the rotation group or the special orthogonal group. It is a subset of the orthogonal group, which includes reflections and consists of all orthogonal matrices with determinant 1 or -1, and of the special linear group, which includes all volume-preserving transformations and consists of matrices with determinant 1.

http://en.wikipedia.org/wiki/Rotation_matrix

As in two dimensions a matrix can be used to rotate a point (x, y, z) to a point (x′, y′, z′). The matrix used is a 3 × 3 matrix,

$\mathbf{A} = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$

This is multiplied by a vector representing the point to give the result

$\mathbf{A} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} x' \\ y' \\ z' \end{pmatrix}$

The matrix A is a member of the three dimensional special orthogonal group, SO(3), that is it is an orthogonal matrix withdeterminant 1. That it is an orthogonal matrix means that its rows are a set of orthogonal unit vectors (so they are an orthonormal basis) as are its columns, making it easy to spot and check if a matrix is a valid rotation matrix. The determinant must be 1 as if it is -1 (the only other possibility for an orthogonal matrix) then the transformation given by it is a reflection, improper rotation or inversion in a point, i.e. not a rotation.

Matrices are often used for doing transformations, especially when a large number of points are being transformed, as they are a direct representation of the linear operator. Rotations represented in other ways are often converted to matrices before being used. They can be extended to represent rotations and transformations at the same time using Homogeneous coordinates. Transformations in this space are represented by 4 × 4 matrices, which are not rotation matrices but which have a 3 × 3 rotation matrix in the upper left corner.

The main disadvantage of matrices is that they are more expensive to calculate and do calculations with. Also in calculations wherenumerical instability is a concern matrices can be more prone to it, so calculations to restore orthonormality, which are expensive to do for matrices, need to be done more often.

Unitary matrix

From Wikipedia, the free encyclopedia

In mathematics, a unitary matrix is an $n\times n$ complex matrix

U

satisfying the condition

$U^{\dagger} U = UU^{\dagger} = I_n\,$

where

I n

is the identity matrix in n dimensions and $U^{\dagger}$ is the conjugate transpose (also called the Hermitian adjoint) of

U

. Note this condition says that a matrix

U

is unitary if and only if it has an inverse which is equal to its conjugate transpose $U^{\dagger} \,$

$U^{-1} = U^{\dagger} \,\;$

A unitary matrix in which all entries are real is an orthogonal matrix. Just as an orthogonal matrix

G

preserves the (real) inner productof two real vectors,

$\langle Gx, Gy \rangle = \langle x, y \rangle$

so also a unitary matrix

U

satisfies

$\langle Ux, Uy \rangle = \langle x, y \rangle$

for all complex vectors x and y, where $\langle\cdot,\cdot\rangle$ stands now for the standard inner product on $\mathbb{C}^n$ .

If $U \,$ is an n by n matrix then the following are all equivalent conditions:

$U \,$ is unitary
$U^{\dagger} \,$ is unitary
the columns of $U \,$ form an orthonormal basis of $\mathbb{C}^n$ with respect to this inner product
the rows of $U \,$ form an orthonormal basis of $\mathbb{C}^n$ with respect to this inner product
$U \,$ is an isometry with respect to the norm from this inner product
$U \,$ is a normal matrix with eigenvalues lying on the unit circle.

http://en.wikipedia.org/wiki/Unitary_matrix

Normal matrix

From Wikipedia, the free encyclopedia

A complex square matrix A is a normal matrix if

$A^*A=AA^* \$

where A* is the conjugate transpose of A. That is, a matrix is normal if it commutes with its conjugate transpose.

If A is a real matrix, then A*=A^T; it is normal if A^TA = AA^T.

Normality is a convenient test for diagonalizability: every normal matrix can be converted to a diagonal matrix by a unitary transform, and every matrix which can be made diagonal by a unitary transform is also normal, but finding the desired transform requires much more work than simply testing to see whether the matrix is normal.

The concept of normal matrices can be extended to normal operators on infinite dimensional Hilbert spaces and to normal elements in C*-algebras. As in the matrix case, normality means commutativity is preserved, to the extent possible, in the noncommutative setting. This makes normal operators, and normal elements of C*-algebras, more amenable to analysis.

http://en.wikipedia.org/wiki/Normal_matrix

Hermitian matrix

From Wikipedia, the free encyclopedia

In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries that is equal to its ownconjugate transpose – that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j:

$a_{i,j} = \overline{a_{j,i}}\,.$

If the conjugate transpose of a matrix

A

is denoted by $A^\dagger$ , then the Hermitian property can be written concisely as

$A = A^\dagger\,.$

Hermitian matrices can be understood as the complex extension of real symmetric matrices.

Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of having eigenvalues always real.

http://en.wikipedia.org/wiki/Hermitian_matrix

算盤

Nube de etiquetas

2011-02-09

Nodal Discontinuos Galerkin methods

2011-02-04

dot product

Vector formulation

cross product

2011-02-01

Rotation matrix

Rotation matrix

Unitary matrix

Unitary matrix

Normal matrix

Normal matrix

Hermitian matrix

Hermitian matrix

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