Study notes : Linear Algebra (Part – I)

 

Type of Matrix

Row Matrix:

A matrix with only one row is called a row matrix.

e.g. =  [ 1    2    4 ]_1×3

Column Matrix:

A matrix with only one column is called a column matrix.

Eg:           1

                2

                3     _3×1

Square Matrix:

A square matrix has an equal number of rows and columns. (i.e.) m = n.

Eg =          1       2     3

                 4       5     6

                 8       9     7    _3×3

 Diagonal Matrix:

A square matrix is called a diagonal matrix if each of its non-diagonal elements are zero  (i.e.) A square matrix A is diagonal if a_ij ≠ 0 for i = j and
a_ij = 0 for i ≠ j.

Eg =          1       0     0

                 0       2     0

                 0       0     3    _3×3

Null Matrix:

It is a matrix whose all elements are 0.

Scalar Matrix:

A diagonal matrix whose diagonal elements are equal is called a scalar matrix.

Eg =          4       0     0

                 0       4     0

                 0       0     4    _3×3

Upper Triangular Matrix:

An upper triangular matrix is one whose entry values below the leading diagonal are zero.

Eg =          5       6     8

                 0       1     2

                 0       0     3    _3×3

Lower Triangular Matrix:

A lower triangular matrix is one whose entry values above the leading diagonal are zeros.

Eg =          5       0     0

                 6       2     0

                 7       3     8    _3×3

Equality of Matrices:

Two matrices are equal if they are of the same dimension and their corresponding entries are equal.

Eg:       A  =  B

Where    A =         1     2               B  =          1     2

                            5     6     _2×2                      5     6     _2×2

Transpose of a Matrix:

A matrix obtained from any given matrix A, by interchanging its rows and columns is called the transpose of A an dis usually denoted by A’ (or) A^T.

Thus if A = [a_ij], then A’ = [b_ij] where b_ij = a_ij.

Eg:           A =        1     2               A’  =         1     4

                             4     5                                2     5  

Conjugate of A Matrix:

The matrix obtained from given matrix A, on replacing its elements by the corresponding conjugate complex numbers is called the conjugate of A and denoted by A.   Thus if A = [a_ij], then A = [b_ij] where b_ij = a_ij  is called the conjugate of A.

Eg:          A  =        1+2i       3-4i             A’  =         1-2i       3+4i

                             1-5i        2-6i                            1+5i      2+6i

Conjugate transpose of A Matrix:

The conjugate of the transpose of a matrix A is called its conjugate transpose and is denoted by A^(H).  Thus A^T.

Eg:          A  =       1+2i       3-4i             A^T =       1-2i       1+5i

                             1-5i        2-6i                            3+4i      2+6i

Symmetric Matrices:

A square matrix A = [a_ij] is said to be symmetric if a_ij = a_ji ,

(i.e.)        the (i,j)^th element is same as the (j, i)^th element.

Eg:          A  =       a      h      g        =       A’ =        a      h      g

                             h     b      f                                h     b      f

                             g      f       c                                g      f       c   .

therefore A = A’.

Skew-Symmetric Matrices:

A square matrix A = [a_ij] is said to be skew-symmetric of a_ij = -a_ji. 

(i.e.)          the (i, j)^th element is the negative of its (j, i)^th element.

Eg:          A  =       0     a      b                 A’ =        0     -a     -b

                            -a    0       c                               a      0      -c

                                _{-b     -c         0                                    b        c          0}  

therefore A = -A’.

Hermitian Matrices:

 A square matrix A = [a_ij] is said to be Hermitian if a_ij = a_ji.

 (i.e.)  the (i, j)^th elements is the conjugate complex of the (j, i)^th elements.

 (i.e.)  for a Hermitian matrix A, we have A = A’.

Skew-Hermitian Matrices:

A square matrix A = [a_ij] is said to be skew-hermitian  if a_ij = -a_ji.

(i.e.)  the (i, j)^th elements is the negative conjugate complex of the (j, i)^th elements.

Eg:      Hermitian matrices                                              Skew-Hermitian Matrices

            A =         1            2-3i                                       A’ =          1           2+3i

                          2+3i       0                                                           2-3i       0

  

            A’ =       1          2+3i                                        -A’ =        -1          -2-3i

                          2-3i        0                                                       -2+3i         0

 

           -A’ =        1            2-3i                                            

                           2-3i         0

Determinant of A matrix:

If  A = [a_ij] be a square matrix of order n, then the determinant of [a_ij] of order n, is called the determinant of the square matrix A, denoted by |A|.

Eg:            

      A=    

  |A| =  1(45-48) – 2(36-42) + 3(32-35)

            =  (-3) – 2(-6) + 3(-3)

            =  (-3) + 12 – 9

            =  -12 + 12

            =  0.

Minors and Cofactors

            The determinant of a square submatrix of a matrix is called a minor of A.

Minors:

            The minor of an element in a determinant is the determinant got by suppressing the row and column in which the element appears.  The order of the minor of an element in determinant A is less than the order of IAI.  The minor of the element in the i^th row.  j^th column is denoted by m_ij [The determinant of a square submatrix of a matrix A is called a minor of A].

Cofactor

The signed minor is called the cofactor.  The cofactor of the element in the i^th row, j^th column is denoted by c_ij and c_ij = (-1)^i+j m_ij.  The expansion of 3^rd order determinant, when expanded in terms of minors, the sign of the cofactor of the element will be as follows.

           

Adjoint or adjugate of a square matrix:

The adjoint or adjugate of a square matrix A is a transpose of the matrix formed by cofactors of elements of IAI Adjoint of the matrix.  A is denoted as adj A

    adj A          

Note:  If A is a non-singular square matrix, then A(adj A) = (adj A) A = IAI I where I am a unit matrix of the same order as A.

Principle minor:

The principal minor of a matrix is the determinant of a square submatrix formed by deleting a corresponding row and column vectors. 

Eg, The principal minors of the 3×3 matrix

    adj A           

order 1 and

         

Of order 2.

3. Operations on a matrix

3.1  Addition

• The order of the matrices must be the same
• Add corresponding elements together
• Matrix addition is commutative
• Matrix addition is associative

A =

1 2 3 7 8 9

B =

6 7 4 5

Both matrices have the same number of rows and columns (2 rows and 3 columns), so they can be added and subtracted. Thus,

A + B =

1 + 5 2 + 6 3 + 7 7 + 3 8 + 4 9 + 5

A+ B=

6 8 10 10 12 14

 

3.2  Subtraction

• The order of the matrices must be the same
• Subtract corresponding elements
• Matrix subtraction is not commutative (neither is the subtraction of real numbers)
• Matrix subtraction is not associative (neither is the subtraction of real numbers)

A =

1 2 3   7 8 9

B =

5 6 7   3 4 5

Both matrices have the same number of rows and columns (2 rows and 3 columns), so they can be added and subtracted. Thus, And,

A – B =

1 – 5 2 – 6 3 – 7   7 – 3 8 – 4 9 – 5

A-B =

-4 -4 -4   4 4 4

 

 3.3  Matrix Multiplication

A_m×n × B_n×p = C_m×p

• The number of columns in the first matrix must be equal to the number of rows in the second matrix. That is, the inner dimensions must be the same.
• The order of the product is the number of rows in the first matrix by the number of columns in the second matrix. That is, the dimensions of the product are the outer dimensions.
• Since the number of columns in the first matrix is equal to the number of rows in the second matrix, you can pair up entries.
• Each element in row i from the first matrix is paired up with an element in column j from the second matrix.
• The product’s element in row i, column j, is formed by multiplying these paired elements and summing them.
• Each element in the product is the sum of the products of the elements from row i of the first matrix and column j of the second matrix.
• There will be n products which are summed for each element in the product.

Example:

Consider the product of a 2×3 matrix and a 3×4 matrix. The multiplication is defined because the inner dimensions (3) are the same. The product will be a 2×4 matrix, the outer dimensions.

 

		Column 1	Column 2	Column 3	Column 4
	values	[1, -3, 6]	[-8, 6, 5]	[4, 7, -1]	[-3, 2, 4]
Row 1	[1, -2, 3]	1(1) – 2(-3) + 3(6) = 1 + 6 + 18 = 25	1(-8) -2(6) + 3(5) = -8 – 12 + 15 = -5	1(4) -2(7) +3(-1) = 4 – 14 – 3 = -13	1(-3) -2(2) + 3(4) = -3 -4 + 12 = 5
Row 2	[4, 5, -2]	4(1) + 5(-3) -2(6) = 4 – 15 – 12 = -23	4(-8) + 5(6) – 2(5) = -32 + 30 – 10 = -12	4(4) + 5(7) -2(-1) = 16 + 35 + 2 = 53	4(-3) + 5(2) -2(4) = -12 + 10 – 8 = -10

 

So, the final product is

25	-5	-13	5
-23	-12	53	-10

 

4. Properties of Matrices

Property	Example
Commutativity of Addition	A + B = B + A
Associativity of Addition	A + ( B + C ) = ( A + B ) + C
Associativity of Scalar Multiplication	(cd) A = c (dA)
Scalar Identity	1A = A(1) = A
Distributive	c (A + B) = cA + cB
Distributive	(c + d) A = cA + dA
Additive Identity	A + O = O + A = A
Associativity of Multiplication	A (BC) = (AB) C
Left Distributive	A (B + C) = AB + AC
Right Distributive	( A + B ) C = AC + BC
Scalar Associativity / Commutativity	c (AB) = (cA) B = A (cB) = (AB) c
Multiplicative Identity	IA = AI = A

Thanks,

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