Matrices, Introduction and different types

A matrix is a rectangular array of numbers.

An array of mn numbers written in m rows and n columns is called a matrix of order m x n.

If in a matrix, m ≠ n, then it is a rectangular matrix.
If in a matrix, m=n, then it is a square matrix.

If a matrix contains only one row, (i.e. 1xn matrix), then it is a row matrix.

If a matrix contains only 1 column, (i.e. mx1 matrix), then it is a column matrix.

A matrix whose every element is zero is a zero matrix.

A square matrix is called a diagonal matrix if all its elements other than the elements in the leading diagonal are zero.

A square matrix whose elements in the leading diagonal are each equal to 1 and all the other elements are zero is a unit matrix.

Equality of matrices

Two matrices A and B of the same order are equal when their corresponding elements are equal.

is equal to

implies a=1, b=2, c=3, d=4, e=5, f=6, g=7, h=8, and i=9.

Singular and nonsingular matrices

A square matrix is a singular matrix if |A|=0. If |A|≠ 0, then the matrix is nonsingular.
|A| represents the value of the determinant of the matrix.

Scalar multiplication of a matrix by a number

If

then

where k is a scalar or a number.

Negative of a matrix

The negative of a matrix is obtained by multiplying all the elements of the matrix by -1.
The negative of

is

Products of Vectors and Matrices

Scalar Product

A vector or a matrix can be multiplied by a scalar k entirely.
For Example

Vector Product

two vectors containing same amount of entries can be multiplied. some people call this "dot product"

or equivalently, using matrix rule of product expression:

notice the second vector is placed verticle in matrix rule of product expression.

We can think the product as each entry of the former vector(a,b and c) , is scalar multiplied by corresponding entry of the latter vector, and then the 3 product ad, be and cf are added up and give the final result. So does the reverse.(this concept will be applied below next *)

Extension to Matrix Vector Product

We can add another vector [a₂ b₂ c₂]under the vector [a b c] and let it do the SAME multiplication to [d e f], and do the SAME summation and the result is placed under previous one for vector[a₂ b₂ c₂] has been placed under[a b c]:

Recall the concept of scaler product-sum analyze, we will notice both a and a₂ are multiplied by scalar d, both b and b₂ are multiplied by e, as well as both c and c₂ are multiplied by f, and then corresponding product are added. If we define

then the matrix product can be expressed as

where col stands for column, A is now a matrix and no longer a vector .

we can add many rows
[a₃ b₃ c₃]... [a_m b_m c_m] to the matrix, but

still holds.
or

This is called a matrix multiplied by a vector on the right is equivalent to get its columns(also vectors) combined by entries of the right vector.It's another way to perceive matrice product.

Similarly, we have this formula about row combination*

Extension to Matrice Product
what if  a matrix multiplied by a matrix? we can either seperate right matrix(B) into columns of scalars and get a Row of Column combinations or seperate left matrix(A) into rows of scalars and get a Column of Row combinations.

Last edited by George,Y (2006-04-20 13:38:19)

Adjoint of a square Matrix

If A is a square matrix

The adjoint of A is defined to be the transpose of the cofactor matrix of A and is denoted by adj.A.

Inverse of a Matrix

The Inverse or Reciprocal of a non-Singular Matrix A  is denoted by

.

It can be shown that

If A and B are two matrices such that
AB=BA=1, then

Symmetric and Skew-symmetric Matrices

A square matrix is said be symmetric if the (I,j)th element of the matrix is equal to the (j,i)th element.

for all values of i and j.

A square matrix is said to be skew symmetric if the (i,j)the element is equal to the negative of the (j,i)th element.

for all values of i and j.

Eamples:-

Symmetric matrices

Skew-symmetric Matrix

Conjugate of a Matrix

The Matrix obtained from any given Matrix A by replacing its elements by the corresponding conujugate complex numbers is called the conjugate of A and denoted by

If

then

Hermitian and skew-Hermitian Matrices

A square matrix

is said to be Hermitian if the (i,j)th element of A is equal to the conjugate complex of the (j,i)th element of A.

A square matrix

is said to be skew-Hermitian if the (i,j)th element of A is equal to the negative of the conjugate complexof the (j,i)th element.

Example :- Hermitian Matrix

Example:- Skew-Hermitian Matrix

Simultaneous Linear Equations

System of Linear equations can be solved with the help of Matrices.

Consider the linear equations

Let

and

Matrix X is the solution of the given simultaneous equations.

Cramer's Rule for solving Simultaneous Equations

If

Let

the system of equations can be solved by this method.

George,Y,

where i=1, j=2 (the 1st row, 2nd column)

The power of (-1) takes care of the symbol, hence,

IS CORRECT!