Linear Algebra-Matrices

Introduction:

Applications of matrices are found in every branch of engineering, Including electromagnetism, quantum mechanics, quantum electrodynamics, classical mechanics, optics and computer graphics.

Matrix:

A set of mn numbers (real or complex) arranged in the form of a rectangular array of m rows and n columns is known as a matrix of order mxn. The matrices are generally denoted by capital letters such as A,B,C etc. The numbers in the matrix are called elements or entries of the matrix.

An mxn matrix is usually written as .

Equality of Two Matrices

Two matrices A = (aij)mxn and B = (Bij)pxq are said to be equal if

i) they are of same order, i.e., m = p, n = q
and

ii) their corresponding elements are equal,

i.e., aij = bij for all i, j.

Types of Matrices:

Real matrix: If all the elements of a matrix A are real numbers then the matrix A is called a real matrix.

Complex matrix:

If at least one of the elements of a matrix A is purely imaginary (or) complex then the matrix A is called complex matrix.

Row matrix:

If a matrix A has only one row and any number of columns then the matrix A is called a row matrix.

Column matrix:

If a matrix A has only one column and any number of rows then the matrix A is called a column matrix.

Square matrix:

If the number of rows is equal to the number of columns in a matrix then the matrix is called square matrix. A square matrix of order nxn is sometimes called as n-rowed matrix A or simply a square matrix of order ‘n’.

Diagonal or principal diagonal elements:

If A = (aij)nxn then the elements aij of a square matrix for which i = j,

i.e., the elements a11, a22, ……….… ann are called diagonal elements (or) leading diagonal elements.

a11=1, a22=5, a33=8 are diagonal elements.

Trace of a matrix:

The sum of the diagonal elements of a square matrix is called trace of A and it is denoted by trace (A) or tr(A)

Thus, if A = (aij)nxn

Then tr(Anxn) = a11+a22+……. +ann =

Properties:

If A and B are square matrices of order n then

(i) tr(AB) = tr(BA)

(ii) tr(AB) tr(A) tr(B)

(iii) tr(BA) tr(B) tr(A)

(iv) tr(A–B) = tr(A) – tr(B)

(v) tr(A+B) = tr(A)+tr(B)

(vi) tr(kA) = k tr(A) where k is a scalar.

Rectangular matrix:

If the number of rows is not equal to the number of columns in a matrix then the matrix is called a rectangular matrix.

Diagonal matrix:

If all the non-diagonal or off-diagonal elements in a square matrix are zero then the matrix is called diagonal matrix.

Note:

1. A matrix A = [aij]nxn is diagonal matrix
if aij =

  1. If A and B are diagonal matrices then
  2. A+B is a diagonal matrix
  3. A–B is a diagonal matrix
  4. Adj(A), adj(B) are diagonal
    matrices
  5. A–1, B–1 are diagonal matrices

Scalar matrix:

If all the diagonal elements of a diagonal matrix are same or equal then the matrix is called a scalar matrix.

Unit matrix or Identity matrix:

If all the diagonal elements of a diagonal matrix are one then the matrix is called an identity matrix.

A unit matrix of order n is denoted by In.

Upper triangular matrix:

A = [aij]nxn is an upper triangular matrix

Triangular matrix:

A matrix which is either upper triangular or lower triangular is called a triangular matrix.

Strictly triangular matrix:

A triangular matrix A = (aij)nxn is called a strictly triangular matrix

If aii = 0 for all i = 1, 2, …… n

Idempotent matrix:

If A2 = A for a square matrix A of order n then Anxn is called an idempotent matrix.

  • If AB = A and BA = B then A and B are idempotent.
  • If AB = BA = 0, then the sum of two idempotent matrices A and B is an
    idempotent.

Nilpotent matrix:

If there exists a positive integer m for a square matrix A of order m such that Am = 0 then the matrix A is called a nilpotent matrix.

If m is a least positive integer for which Am = 0 then ‘m’ is called the index of the nilpotent matrix.

a nilpotent
matrix of index 2.

Involuntary matrix:

If A2 = I for a square matrix A then matrix A is called an involuntary.

Transpose of a matrix:

If a matrix Bnxm is obtained from a matrix Amxn by changing its rows into columns and its columns into rows then the matrix Bnxm is called transpose of A and is denoted by AT or A1.

Thus, if A = (aij)mxn then AT = [aji]nxm

Properties of Transpose:

If A and B are two matrices and AT and BT are transpose of A and B respectively then

(i) (AT)T = A

(ii) (AB)T = BTAT

(iii) (kA)T = kAT where k is a scalar (real or
complex)

(iv) (AB)T = ATBT

Symmetric matrix:

If aij = aji for all i, j in A = (aij)nxn then Anxn is called a Symmetric matrix.

i.e. AT = A

Skew-symmetric matrix:

A square matrix A is said to be Skew-symmetric matrix if AT =  –A.

A = (aij)nxn is a Skew-symmetric matrix

then aij = –aji­  i,  j

Properties of Symmetric & Skew-symmetric matrices:

If A and AT are any two square matrices then

  • A + AT is symmetric
  • A – AT , AT – A are skew-symmetric
  • AAT, ATA are symmetric

If A and B are symmetric matrices then

(i)   A B are symmetric

(ii)  AB, BA are not symmetric

(iii) AB + BA is symmetric

(iv) AB – BA is skew-symmetric

If A and B are skew-symmetric then

(i)   A B are skew-symmetric

(ii)  AB, BA are not skew-symmetric

(iii) A2, B2 are symmetric

(iv) A2 B2 are symmetric

  • Every square matrix A can be uniquely expressed as a sum of symmetric and Skew-symmetric matrices.
  • If A and B are square symmetric matrices then AB is symmetric
  • If A and B are skew-symmetric matrices then AB is symmetric A and B are commute

 

 

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