Vectors and Matrices

 
A vector, u, means a list (or n-tuple) of numbers:
u = (u1, u2, . . . , un)

where ui are called the components of u. If all the ui are zero i.e., ui = 0, then u is called the zero vector.

Given vectors u and v are equal i.e., u = v, if they have the same number of components and if corresponding components are equal.

Addition of Two Vectors

If two vectors, u and v, have the number of components, their sum, u + v, is the vector obtained by adding corresponding components from u and v.
        u + v = (u1, u2, . . . , un) + (v1, v2, . . . , vn)
                 = (u1 + v1 + u2 + v2, . . . , un + vn)


Multiplication of a vector by a Scalar

The product of a scalar k and a vector u i.e., ku, is the vector obtained by multiplying each component of u by k:
        ku  = k(u1, u2, . . . , un)
             = ku1, ku2, . . . , kun

 Here, we define        -u = (-1)u      and        u-v = u +(-v)



It is not difficult to see k(u + v) = ku + kv where k is a scalar and u and v are vectors

Dot Product and Norm

The dot product or inner product of vectors u = (u1, u2, . . . , un) and v = (v1, v2, . . . , vn) is denoted by u.v and defined by
u.v = u1v1 + u2v2 + . . . + unvn

The norm or length of a vector, u, is denoted by ||u|| and defined by

      


Matrices

Matrix, A, means a rectangular array of numbers
       A =

The m horizontal n-tuples are called the rows of A, and the n vertical m-tuples, its columns. Note that the element aij, called the ij-entry, appear in the ith row and the jth column.

In algorithmic (study of algorithms), we like to write a matrix A as A(aij).

Column Vector

A matrix with only one column is called a column vector

Zero Matrix

A matrix whose entries are all zero is called a zero matrix and denoted by 0.

Matrix Addition

Let A and B be two matrices of the same size. The sum of A and B is written as A + B and obtained by adding corresponding elements from A and B.


         A+B =



                =


Scalar Multiplication

The product of a scalar k and a matrix A, written kA or Ak, is the matrix obtained by multiplying each element of A by k.
        kA  = k


               =  


Here, we define     -A = (-1)A        and        A - B = A + (-B). Note that -A is the negative of the matrix A.



Properties of Matrix under Addition and Multiplication

Let A, B, and C be matrices of same size and let k and l be scalars. Then
  1. (A + B) + A + (B + C)
  2. A + B = B + A
  3. A + 0 = 0 + A = A
  4. A + (-A) = (-A) + A = 0
  5. k(A + B) = kA + kB
  6. (k + l)A = kA + lA
  7. (kl)A = k(lA)
  8. lA = A

Matrix Multiplication

Suppose A and B are two matrices such that the number of columns of A is equal to number of rows of B. Say matrix A is an m×p matrix and matrix B is a p×n matrix. Then the product of A and B is the m×n matrix whose ij-entry is obtained by multiplying the elements of the ith row of a by the corresponding elements of the jth column of B and then adding them.

It is important to note that if the number of columns of A is not equal to the number of rows of B, then the product AB is not defined.


Properties of Matrix Multiplication

Let A, B, and C be matrices and let k be a scalar. Then
  1. (AB)C = A(BC)
  2. A(B+C) = AB + AC
  3. (B+C)A = BA + CA
  4. k(AB) = (kB)B = A(kB)


Transpose

The transpose of a matrix A is obtained by writing the row of A, in order, as columns and denoted by AT. In other words, if A - (Aij), then B = (bij) is the transpose of A if bij - aji for all i and j.

It is not hard to see that if A is an m×n matrix, then AT is an n×m matrix.

For example if  A = , then AT =


Square Matrix

If  the number of rows and the number of columns of any matrix are the same, we say matrix is a square matrix, i.e., a square matrix has same number of rows and columns. A square matrix with n rows and n columns is said to be order n and is called an n-square matrix.

The main diagonal, or simply diagonal, of an n-square matrix A = (aij) consists of  the elements a11, a22, a33, . . . , amn.


Unit Matrix

The n-square matrix with 1's along the main diagonal and 0's elsewhere is called the unit matrix and usually denoted by I.

Why unit matrix?
The unit matrix plays the same role in matrix multiplication as the number 1 does in the usual multiplication of numbers.
            AI = IA = A
for any square matrix A.


So what dude?
We can form powers of a square matrix X by defining

X2 = XX
X3 = X2X, XXX, . . .
X0 = I

Big deal!
We can also form polynomial in X. That is, for any polynomial

f(x) = a0 + a1x + a2x2 + . . . + anxn
We define f(x) to be the matrix
f(x) = a0I + a1x + a2x2 + . . . + anxn
Note that in the case that f(A) is the zero matrix, then matrix A said to be a zero of the polynomial f(x) or a root of the polynomial equation f(x) = 0.


Now you're talking !!!!


Invertible Matrices

A square  matrix  A is said to be invertible if there exists a matrix B with the property AB = BA = I (Identity Matrix). Such a matrix B is unique and it is called the matrix of A and is denoted by A-1. Here, the important observation is that B is the inverse of A if and only if A is the matrix of B. It is known that AB = I if and only if BA = I; hence it is necessary to test only one product to determine whether two given matrices are inverse.

Determinants

To each n-square matrix A = (aij), we assign a specific number called the determinants of A and denoted as
|A|     =     del(A)

 =  

where an n by n arrays of numbers enclosed by straight lines is called a determinant of order n.

It is important to note that n by n array of numbers enclosed by straight line (see above) is NOT a matrix but denotes the number that the determinant function assigns to the enclosed array of numbers, i.e., the enclosed square matrix.

The determinant of order one is  |a11| = a11

The determinant of order two is   = a11a22 - a12a21

The determinant of order three is    = a11a22a32+ a12a23a31 + a13a21a32 - a13a22a31 - a12a21a33 - a11a23a32

The determinant of order fo... You get the picture !


General Definition of Determinant

The general definition of  a determinant of order n is as follows
det(A) =

where the sum is over all possible permutations sigma = (j1, j2 , . . . , jn ) of (1, 2, . . . , n). Here sgn(sigma) equals plus or minus one according as an even or an odd number of interchanges are required to change sigma so that its numbers are in the usual order.


An important property of the determinant function is

Lemma. For any two n-square matrices A and B, det(AB) = det(A) . det(B).

In simple words the lemma say that the determinant function is multiplicative.


An important point in the context of invertible matrices and determinant is

Lemma. A square matrix is invertible if and only if it has a nonzero determinant.


A good news and a bad news: If  you're an under graduate, you're done here! now goto CLR- If you're graduate and interested in theoretical Computer Science its time to go and find some good on matrix theory. (You'll need this shit for Linear Programming, for example).