A vector, u, means a list (or n-tuple) of numbers:

u = (u_{1},
u_{2}, . . . , u_{n})

where u

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.

u + v = (u_{1}, u_{2},
. . . , u_{n}) + (v_{1}, v_{2}, . . . , v_{n})

= (u_{1} + v_{1} + u_{2}
+ v_{2}, . . . , u_{n} + v_{n})

ku
= k(u_{1}, u_{2},
. . . , u_{n})

= ku_{1},
ku_{2}, . . . , ku_{n}

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 = (u

u.v = u_{1}v_{1}
+ u_{2}v_{2} + . . . +
u_{n}v_{n}

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

A =

The m horizontal n-tuples are called the rows of A, and the n vertical m-tuples, its columns. Note that the element a

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

A matrix with only one column is called a column
vector

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

A+B =

=

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

- (A + B) + A + (B + C)
- A + B = B + A
- A + 0 = 0 + A = A
- A + (-A) = (-A) + A = 0
- k(A + B) = kA + kB
- (k + l)A = kA + lA
- (kl)A = k(lA)
- lA = A

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

- (AB)C = A(BC)
- A(B+C) = AB + AC
- (B+C)A = BA + CA
- k(AB) = (kB)B = A(kB)

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

For example if A = , then A

The main diagonal, or simply diagonal, of an n-square matrix A = (a

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

X^{2} = XX

X^{3} = X^{2}X,
XXX, . . .

X^{0} = I

Big deal!

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

f(x) = a_{0}
+ a_{1}x + a_{2}x^{2} + .
. . + a_{n}x^{n}

We define f(x) to be
the matrix

f(x) = a_{0}I
+ a_{1}x + a_{2}x^{2} + .
. . + a_{n}x^{n}

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 !!!!

|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 |a

The determinant of order two is = a

det(A) =

where the sum is over all possible permutations = (j

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).