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the Gram–Schmidt process

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In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method fororthonormalising a set of vectors in an inner product space, most commonly the Euclidean space Rn. The Gram–Schmidt process takes a finite, linearly independent set S = {v1, …, vk} for k ≤ n and generates an orthogonal setS′ = {u1, …, uk} that spans the same k-dimensional subspace of Rn as S.

The method is named after Jørgen Pedersen Gram and Erhard Schmidt but it appeared earlier in the work of Laplaceand Cauchy. In the theory of Lie group decompositions it is generalized by the Iwasawa decomposition.[1]

The application of the Gram–Schmidt process to the column vectors of a full column rank matrix yields the QR decomposition (it is decomposed into an orthogonal and a triangular matrix).

In linear algebra, a QR decomposition (also called a QR factorization) of a matrix is a decomposition of a matrix A into a product A = QR of an orthogonal matrix Q and anupper triangular matrix R. QR decomposition is often used to solve the linear least squares problem, and is the basis for a particular eigenvalue algorithm, the QR algorithm.

If A has n linearly independent columns, then the first n columns of Q form an orthonormal basis for the column space of A. More specifically, the first k columns of Q form an orthonormal basis for the span of the first k columns of A for any 1 ≤ k ≤ n.[1] The fact that any column k of A only depends on the first k columns of Q is responsible for the triangular form of R

 

Unit Vectors in Curvilinear Coordinates

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Swapnil Sunil Jain

Let (u,v,w) be any non-cartesian coordinate system such that x=x(u,v,w), y =y(u,v,w), z=z(u,v,w).

We can combine the above three equations into a single vector equation that gives the position vector {\overrightarrow{\bf r}} of any point P(x,y,z) in space as a function of the coordinates u,v,w:

\displaystyle {\overrightarrow{\bf r}}=x(u,v,w)\hat{\bf i}+y(u,v,w)\hat{\bf j}+z(u,v,w)\hat{\bf k}

If we held $ u$ fixed s.t. $ u=u_0$ then the position vector becomes the parametric equation of the surface (called the coordinate surface) $ u=u_0$ where $ v,w$ play the role of parameters. Furthermore, if we held both $ u$ and $ v$ fixed s.t $ u=u_0$ and $ v=v_0$, then the position vector becomes the parametric equation of the curve (called the coordinate curve) formed by the intersection of the surfaces $ u=u_0$ and $ v=v_0$, in which $ w$ acts as a parameter along the curve.

Now, how do we find the tangent vectors? Well, what is the meaning of a tangent vector? A tangent vector is a vector which is tangent to a coordinate curve formed by the intersection of the two coordinate surfaces. In other words, it is a vector which indicates the direction in which one of the coordinates, say $ u$, increases while the other two coordinates (i.e. $ v$ and $ w$) are held fixed. Sound familiar? Yes, of course, partial derivatives! A partial derivative with respect to $ u$ would take the derivative of the position vector $ \vec{r}$ along the coordinate curve formed by the intersection of the surfaces $ v=v_0$ and $ w=w_0$ and hence return you a tangent vector along that curve. Hence, by taking the partial derivative of $ \vec{r}$ one by one with respect to all three coordinates, we would get all the three tangent vectors which are tangent to their respective coordinate curves. Thus, we arrive at the following three tangent vectors:

\displaystyle \overrightarrow{\bf v_\alpha}=\frac{\partial \overrightarrow{\bf r}}{\partial \alpha}, \alpha = u, v, w .

However, these are not normalized vectors. Most often we are interested in unit tangent vectors. So we divide them by their respective lengths. Therefore,

\displaystyle \overrightarrow{\bf e_\alpha}=\frac{\frac{\partial \overrightarrow{\bf r}}{\partial \alpha}}{|\frac{\partial \overrightarrow{\bf r}}{\partial \alpha}|}, \alpha = u, v, w .

or

\displaystyle \overrightarrow{\bf e_\alpha}=\frac{\frac{\partial \overrightarrow{\bf r}}{\partial \alpha}}{h_\alpha}, \alpha = u, v, w .

where

\displaystyle h_\alpha=\sqrt{{\frac{\partial x}{\partial \alpha}}^2+{\frac{\partial y}{\partial \alpha}}^2+{\frac{\partial z}{\partial \alpha}}^2}

are known as scale or metric factors (or coefficients).