Difference between revisions of "Linear Algebra"

Revision as of 17:13, 25 August 2008

Basics

Let's first make some definitions:

Vector: A one-dimensional collection of numbers. Examples:; $a=\left( \begin{array}{cccc} 1 & 2 & 3 & 4 \end{array} \right), \quad b=\left( \begin{array}{c} 1 \\ 2 \\ 3 \\ 4 \end{array} \right)$

Matrix: A two-dimensional collection of numbers. Examples:; $A = \left[ \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array} \right], \quad B = \left[ \begin{array}{cc} 9 & 6 \\ 2 & 3 \end{array}\right]$

Array: An n-dimensional collection of numbers. A vector is a 1-dimensional array while a matrix is a 2-dimensional array.

Transpose: An operation that involves interchanging the rows and columns of an array. It is indicated by a $^\mathsf{T}$ superscript. For example:; $a=\left( \begin{array}{cccc} 1 & 2 & 3 & 4 \end{array} \right) \; \Rightarrow \; a^\mathsf{T} = \left( \begin{array}{c} 1 \\ 2 \\ 3 \\ 4 \end{array} \right)$; $A = \left[ \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array} \right] \; \Rightarrow \; A^\mathsf{T} = \left[ \begin{array}{cc} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{array} \right]$

Vector Magnitude, $\left|\vec{a}\right|$: A measure of the length of a vector, $\left|\vec{a}\right| = \sqrt{ \sum_{i=1}^{n} a_i^2 }$; Example: $\vec{a}=\left( \begin{array}{ccc} 2 & 8 & 3 \end{array} \right) \; \Rightarrow \; \left|\vec{a}\right| = \sqrt{ 2^2 + 8^2 + 3^2 }$; Example: $\vec{b}=\left( \begin{array}{cccc} b_1 & b_2 & b_3 & b_4\end{array} \right) \; \Rightarrow \; \left|\vec{b}\right| = \sqrt{ b_1^2 + b_2^2 + b_3^2 +b_4^2 }$

Matrix & Vector Algebra

There are many common operations involving matrices and vectors including:

Vector dot product
Vector cross product
Matrix-vector product
Matrix-matrix product

These are each discussed in the following sub-sections.

Vector Dot Product

The dot product of two vectors produces a scalar. Physically, the dot product of a and b represents the projection of a onto b.

Given two vectors a and b, their dot product is formed as

c = \vec{a} \cdot \vec{b} = \sum_{i=1}^n a_i b_i,

where $a_i$ and $b_i$ are the components in vectors a and b respectively. This is most useful when we know the components of the vectors a and b.

Occasionally we know the magnitude of the two vectors and the angle between them. In this case, we can calculate the dot product as

c = |\vec{a}|\,|\vec{b}|\,\cos(\theta),

where θ is the angle between the vectors a and b.

Note that if a and b are both column vectors, then $\vec{a}\cdot \vec{b} = a^\mathsf{T}b$ .

For more information on the dot product, see wikipedia's article.

Dot Product in MATLAB

There are several ways to calculate the dot product of two vectors in MATLAB:

  c = dot(a,b);
  c = a' * b;    % if a and b are both column vectors.

Thought examples

Consider the cartoon shown to the right.

At noon, when the sun is directly overhead, a rocket is launched directly toward the sun at 1000 mph. How fast is its shadow moving?

Define the rocket's velocity as

\vec{R}

.

Define the unit normal on the ground (i.e. the direction of the rocket's shadow on the ground) as

\vec{s}

, where

|\vec{s}| = 1

.

Intuition tells us that if the it is moving directly toward the sun, then its shadow does not appear to move at all. : The dot product

\vec{R}\cdot\vec{s} = |\vec{R}| \, |\vec{s}| \cos(\theta) = 0

since cos(90°)=0.

Consider the same rocket going parallel to the earth's surface. How fast is its shadow moving?

If the rocket is going parallel to the earth's surface, our intuition tells us that is shadow is moving at the same speed. This is confirmed by the mathematics, since the angle between the rocket's path and the ground is 0. Therefore,

\vec{R}\cdot\vec{s} = |\vec{R}| \, |\vec{s}| \cos(\theta) = 0

since cos(0°)=1.

What if the rocket were going at a 45° angle?

Our intuition tells us that the shadow will appear to move, but it will not be moving as fast as the rocket is moving. Mathematically, we have

v_s = \vec{R}\cdot\vec{s} = |\vec{R}| |\vec{s}| \cos(45^\circ) = 500\sqrt{2} = 707.1 \, \textrm{mph}

.

Vector Cross Product

The cross product of two vectors produces a vector perpendicular to the original two. The common right-hand rule is used to determine the direction of the resulting vector.

Assume that we have vectors a and b defined as

$\begin{align} \vec{a} &= a_x \hat{i} + a_y \hat{j} + a_z \hat{k} \\ \vec{b} &= b_x \hat{i} + b_y \hat{j} + b_z \hat{k} \end{align}$

where $\hat{i}$ , $\hat{j}$ , and $\hat{k}$ represent the unit-normal vectors in the x, y, and z directions, respectively. The cross product of a and b is then defined as

\vec{c} = \vec{a} \times \vec{b} = \left( a_y b_z -a_z b_y \right) \hat{i} + \left(a_z b_x -a_x b_z\right) \hat{j} + \left(a_x b_y -a_y b_x\right) \hat{k}

If we know the magnitude of a and b and the angle between them (θ), then the cross-product is given as

\vec{a}\times\vec{b} = |\vec{a}|\,|\vec{b}|\,\sin(\theta) \, \vec{n},

where n is the unit-normal vector perpendicular to the plane defined by the vectors a and b.

For more information, see wikipedia's article on the cross product.

Vector Cross Product in MATLAB

   c = cross(a,b);  % c is a vector!

Matrix-Vector Product

The matrix-vector product is frequently encountered in solving linear systems of equations, although there are many other uses for it.

Consider a matrix A with m rows and n columns and vector b with n rows:

A=\left[\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ a_{m1} & a_{m2} & \cdots & a_{mn}\end{array}\right], \quad b= \left(\begin{array}{c} b_{1}\\ b_{2}\\ \vdots\\ b_{n} \end{array}\right)

We may multiply A and b if the number of columns in A is equal to the number of rows in b. The product, c is a vector with m rows. The i^th entry in c is given as

c_i=\sum_{j=1}^n A_{i,j}b_j

For example, let's consider the following case:

A = \left[\begin{array}{ccc} 1 & 3 & 2 \\ 2 & 5 & 4 \end{array}\right], \quad b = \left(\begin{array}{c} 10 \\ 4 \\ 6 \end{array} \right)

Here we have m=2 and n=3. We define the product, c=Ab, as follows:

Be sure that A and b are compatible for multiplication.
- Since A has three columns and b has three rows, they are compatible.
Determine how many entries are in c.
- The c vector will have as many rows as A has.
- Since A has two rows, c will have two rows.
Determine the elements in c
- Applying the above formula for i=1 we find $\begin{align} c_1 &= \sum_{j=1}^{3} A_{1,j} b_j \\ &= A_{1,1} b_1 + A_{1,2} b_2 + A_{1,3} b_3 \\ &= 1 \cdot 10 + 3\cdot 4 + 2 \cdot 6 \\ &= 34. \end{align}$
- Applying the above formula for i=2 we find $\begin{align} c_2 &= \sum_{j=1}^{3} A_{2,j} b_j \\ &= A_{2,1} b_1 + A_{2,2} b_2 + A_{2,3} b_3 \\ &= 2 \cdot 10 + 4\cdot 4 + 4 \cdot 6 \\ &= 64. \end{align}$

Therefore, we have

c=\left(\begin{array}{c} 34 \\ 64 \end{array}\right).

Matrix-Vector product in MATLAB

   c = A*b;   % A is a matrix, b and c are vectors.

Matrix-Matrix Product

The matrix-matrix product is a generalization of the matrix-vector product. Consider the general case where we want to multiply two matrices, C = A B, where A and B are matrices. The following rules apply:

The number of columns in A must be equal to the number of rows in B.
The number of columns in C is equal to the number of columns in B.
The number of rows in C is equal to the number of rows in A.

We can state this differently. Suppose that A has m rows and p columns, and that B has p rows and n columns. This satisfies rule 1 above. Then, according to rules 2 and 3, C has m rows and n columns.

The general formula for matrix multiplication is

$C_{i,j} = \sum_{k=1}^n A_{i,k} B_{k,j}$

Let's show how this works for a simple example:

A = \left[ \begin{array}{cc} 1 & 2 \\ 3 & 5 \end{array} \right], \quad B = \left[ \begin{array}{ccc} 2 & 4 & 10 \\ 6 & 3 & 1 \end{array} \right].

We can form C=A B since rule 1 is satisfied. From the size of A and B we define m=2 and n=3 and from rules 2 and 3, we conclude that the resulting matrix, C has 2 rows and 3 columns.

We are now ready to build each element of C.

i=1 and j=1: $C_{1,1} = \sum_{k=1}^2 A_{1,k} B_{k,1} = A_{1,1} B_{1,1} + A_{1,2} B_{2,1} = 1 \cdot 2 + 2 \cdot 6 = 14.$

i=2 and j=1: $C_{2,1} = \sum_{k=1}^2 A_{2,k} B_{k,1} = A_{2,1} B_{1,1} + A_{2,2} B_{2,1} = 3 \cdot 2 + 5 \cdot 6 = 36.$
i=1 and j=2: $C_{1,2} = \sum_{k=1}^2 A_{1,k} B_{k,2} = A_{1,1} B_{1,2} + A_{1,2} B_{2,2} = 1 \cdot 4 + 2 \cdot 3 = 10.$
i=2 and j=2: $C_{2,2} = \sum_{k=1}^2 A_{2,k} B_{k,2} = A_{2,1} B_{1,2} + A_{2,2} B_{2,2} = 3 \cdot 4 + 5 \cdot 3 = 27.$
i=1 and j=3: $C_{1,3} = \sum_{k=1}^2 A_{1,k} B_{k,3} = A_{1,1} B_{1,3} + A_{1,2} B_{2,3} = 1 \cdot 10 + 2 \cdot 1 = 12.$
i=2 and j=3: $C_{2,3} = \sum_{k=1}^2 A_{2,k} B_{k,3} = A_{2,1} B_{1,3} + A_{2,2} B_{2,3} = 3 \cdot 10 + 5 \cdot 1 = 35.$

Now, putting all of these together, we have

C=\left[\begin{array}{ccc} 14 & 10 & 12 \\ 36 & 27 & 35 \end{array}\right].

Matrix-Matrix product in MATLAB

   C = A*B;   % A, B, and C are all matrices.

The MATLAB code to implement the above example is

   A = [ 1 2; 3 5; ];
   B = [ 2 4 10; 6 3 1; ];
   C = A*B;

Linear Systems of Equations

Any set of linear systems may be written in the form $[A](x)=(b)$ , where $[A]$ is a matrix and $(x)$ and $(b)$ are vectors. In general, for a system of equations with n unknowns x₁ ... x_n, can be written as

\begin{align} a_{1,1} x_1 + a_{1,2} x_2 + a_{1,3} x_3 + \cdots + a_{1,n} x_n &= b_1, \\ a_{2,1} x_1 + a_{2,2} x_2 + a_{2,3} x_3 + \cdots + a_{2,n} x_n &= b_2, \\ &\vdots \\ a_{n,1} x_1 + a_{n,2} x_2 + a_{n,3} x_3 + \cdots + a_{n,n} x_n &= b_n, \\ \end{align}

where $a_{i,j}$ and $b_i$ are numbers and $x_i$ is the set of unknown variables. These equations can now be written in matrix format:

\left[ \begin{array}{ccccc} a_{1,1} & a_{1,2} & a_{1,3} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & a_{2,3} & \cdots & a_{2,n} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{n,1} & a_{n,2} & a_{n,3} & \cdots & a_{n,n} \end{array} \right] \left( \begin{array}{c} x_1 \\ x_2 \\ \vdots \\ x_n \end{array} \right) = \left( \begin{array}{c} b_1 \\ b_2 \\ \vdots \\ b_n \end{array} \right).

Example

Consider the following equations:

\begin{cases} x + y -5z = 9 \\ 6-z = 2y \\ y-3x = 10z \\ \end{cases}

To put these in matrix format, we first choose the solution vector. Let's choose to order this as x, y, z. Now we rewrite the original system so that we have a coefficient multiplying each of these variables, and then we put the "constant" on the right-hand-side:

\begin{align} 1x + 1y -5z &= 9 \\ 0x - 2y -1z &= -6 \\ -3x + 1y -10z &= 0 \end{align}

We may then identify the A matrix and b vectors and write

\left[\begin{array}{ccc} 1 & 1 & -5 \\ 0 & -2 & -1 \\ -3 & 1 & -10 \\ \end{array} \right] \left( \begin{array}{c} x \\ y \\ z \end{array}\right) = \left( \begin{array}{c} 9 \\ -6 \\ 0 \end{array}\right)

Note that if we multiply the above matrix and vector, we recover the original set of equations (try it to be sure that you can do it).

Note that if we chose a different variable ordering, then it simply rearranges the columns in the matrix. For example, if we order the variables z, x, y then we have

\begin{align} -5z + 1x + 1y &= 9 \\ -1z + 0x - 2y &= -6 \\ -10z - 3x + 1y &= 0 \end{align} \quad \Rightarrow \quad \left[\begin{array}{ccc} -5 & 1 & 1 \\ -1 & 0 & -2 \\ -10 & -3 & 1 \\ \end{array} \right] \left( \begin{array}{c} z \\ x \\ y \end{array}\right) = \left( \begin{array}{c} 9 \\ -6 \\ 0 \end{array}\right)

Solving Linear Systems of Equations

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     C = A*B;
 </source>
 == Linear Systems of Equations ==

Difference between revisions of "Linear Algebra"

Revision as of 17:13, 25 August 2008

Contents

Basics

Matrix & Vector Algebra

Vector Dot Product

Dot Product in MATLAB

Thought examples

Vector Cross Product

Vector Cross Product in MATLAB

Matrix-Vector Product

Matrix-Vector product in MATLAB

Matrix-Matrix Product

Matrix-Matrix product in MATLAB

Linear Systems of Equations

Example

Solving Linear Systems of Equations

Direct Solvers

Gaussian Elimination

The Thomas Algorithm (Tridiagonal Systems)

Iterative Solvers

Jacobi Method

Gauss-Seidel Method

Other Methods

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