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Page 1
LINEAR ALGEBRA
MATRICES
A matrix is a rectangular array of numbers (or functions) enclosed in brackets. These numbers
(or function) are called entries of elements of the matrix.
Example:
??
??
??
2 0.4 8
5 -32 0
order = 2 x 3, 2 = no. of rows, 3 = no. of columns
Special Type of Matrices
1. Square Matrix
A m x n matrix is called as a square matrix if m = n i.e, no of rows = no. of columns
The elements
ij
a when i = j
? ?
11 22
a a ......... are called diagonal elements
Example:
??
??
??
12
45
2. Diagonal Matrix
A square matrix in which all non-diagonal elements are zero and diagonal elements may or
may not be zero.
Example:
??
??
??
10
05
Properties
a. diag [x, y, z] + diag [p, q, r] = diag [x + p, y + q, z + r]
b. diag [x, y, z] × diag [p, q, r] = diag [xp, yq, zr]
c.
? ?
?
??
?? ?
?? ??
??
1
1 1 1
diag x, y, z diag , ,
x y z
d.
? ?
??
??
t
diag x, y, z = diag [x, y, z]
e.
? ?
?? ?? ?
?? ??
n
n n n
diag x, y, z diag x ,y ,z
f. Eigen value of diag [x, y, z] = x, y & z
g. Determinant of diag [x, y, z] = xyz
3. Scalar Matrix
A diagonal matrix in which all diagonal elements are equal.
Page 2
LINEAR ALGEBRA
MATRICES
A matrix is a rectangular array of numbers (or functions) enclosed in brackets. These numbers
(or function) are called entries of elements of the matrix.
Example:
??
??
??
2 0.4 8
5 -32 0
order = 2 x 3, 2 = no. of rows, 3 = no. of columns
Special Type of Matrices
1. Square Matrix
A m x n matrix is called as a square matrix if m = n i.e, no of rows = no. of columns
The elements
ij
a when i = j
? ?
11 22
a a ......... are called diagonal elements
Example:
??
??
??
12
45
2. Diagonal Matrix
A square matrix in which all non-diagonal elements are zero and diagonal elements may or
may not be zero.
Example:
??
??
??
10
05
Properties
a. diag [x, y, z] + diag [p, q, r] = diag [x + p, y + q, z + r]
b. diag [x, y, z] × diag [p, q, r] = diag [xp, yq, zr]
c.
? ?
?
??
?? ?
?? ??
??
1
1 1 1
diag x, y, z diag , ,
x y z
d.
? ?
??
??
t
diag x, y, z = diag [x, y, z]
e.
? ?
?? ?? ?
?? ??
n
n n n
diag x, y, z diag x ,y ,z
f. Eigen value of diag [x, y, z] = x, y & z
g. Determinant of diag [x, y, z] = xyz
3. Scalar Matrix
A diagonal matrix in which all diagonal elements are equal.
4. Identity Matrix
A diagonal matrix whose all diagonal elements are 1. Denoted by I
Properties
a. AI = IA = A
b.
?
n
II
c.
?
?
1
II
d. det(I) = 1
5. Null matrix
An m x n matrix whose all elements are zero. Denoted by O.
Properties:
a. A + O = O + A = A
b. A + (- A) = O
6. Upper Triangular Matrix
A square matrix whose lower off diagonal elements are zero.
Example:
??
??
??
??
??
3 4 5
0 6 7
0 0 9
7. Lower Triangular Matrix
A square matrix whose upper off diagonal elements are zero.
Example:
??
??
??
??
??
3 0 0
4 6 0
5 7 9
8. Idempotent Matrix
A matrix is called Idempotent if ?
2
AA
Example:
??
??
??
10
01
9. Involutary Matrix
A matrix is called Involutary if ?
2
AI .
Page 3
LINEAR ALGEBRA
MATRICES
A matrix is a rectangular array of numbers (or functions) enclosed in brackets. These numbers
(or function) are called entries of elements of the matrix.
Example:
??
??
??
2 0.4 8
5 -32 0
order = 2 x 3, 2 = no. of rows, 3 = no. of columns
Special Type of Matrices
1. Square Matrix
A m x n matrix is called as a square matrix if m = n i.e, no of rows = no. of columns
The elements
ij
a when i = j
? ?
11 22
a a ......... are called diagonal elements
Example:
??
??
??
12
45
2. Diagonal Matrix
A square matrix in which all non-diagonal elements are zero and diagonal elements may or
may not be zero.
Example:
??
??
??
10
05
Properties
a. diag [x, y, z] + diag [p, q, r] = diag [x + p, y + q, z + r]
b. diag [x, y, z] × diag [p, q, r] = diag [xp, yq, zr]
c.
? ?
?
??
?? ?
?? ??
??
1
1 1 1
diag x, y, z diag , ,
x y z
d.
? ?
??
??
t
diag x, y, z = diag [x, y, z]
e.
? ?
?? ?? ?
?? ??
n
n n n
diag x, y, z diag x ,y ,z
f. Eigen value of diag [x, y, z] = x, y & z
g. Determinant of diag [x, y, z] = xyz
3. Scalar Matrix
A diagonal matrix in which all diagonal elements are equal.
4. Identity Matrix
A diagonal matrix whose all diagonal elements are 1. Denoted by I
Properties
a. AI = IA = A
b.
?
n
II
c.
?
?
1
II
d. det(I) = 1
5. Null matrix
An m x n matrix whose all elements are zero. Denoted by O.
Properties:
a. A + O = O + A = A
b. A + (- A) = O
6. Upper Triangular Matrix
A square matrix whose lower off diagonal elements are zero.
Example:
??
??
??
??
??
3 4 5
0 6 7
0 0 9
7. Lower Triangular Matrix
A square matrix whose upper off diagonal elements are zero.
Example:
??
??
??
??
??
3 0 0
4 6 0
5 7 9
8. Idempotent Matrix
A matrix is called Idempotent if ?
2
AA
Example:
??
??
??
10
01
9. Involutary Matrix
A matrix is called Involutary if ?
2
AI .
Matrix Equality
Two matrices ? ?
? mn
A and ? ?
? p q
B are equal if
m = p ; n = q i.e., both have same size
ij
a =
ij
b for all values of i & j.
Addition of Matrices
For addition to be performed, the size of both matrices should be same.
If [C] = [A] + [B]
Then ??
ij ij ij
c a b
i.e., elements in same position in the two matrices are added.
Subtraction of Matrices
[C] = [A] – [B]
= [A] + [–B]
Difference is obtained by subtraction of all elements of B from elements of A.
Hence here also, same size matrices should be there.
Scalar Multiplication
The product of any m × n matrix A
??
??
??
jk
a and any scalar c, written as cA, is the m × n
matrix cA =
??
??
??
jk
ca obtained by multiplying each entry in A by c.
Multiplication of two matrices
Let ? ?
? m n
A and ? ?
? p q
B be two matrices and C = AB, then for multiplication, [n = p]
should hold. Then,
?
?
?
n
jk
j1
ik ij
C a b
Properties
? If AB exists then BA does not necessarily exists.
Example: ? ?
? 3 4
A , ? ?
? 4 5
B , then AB exits but BA does not exists as 5 ? 3
So, matrix multiplication is not commutative.
Page 4
LINEAR ALGEBRA
MATRICES
A matrix is a rectangular array of numbers (or functions) enclosed in brackets. These numbers
(or function) are called entries of elements of the matrix.
Example:
??
??
??
2 0.4 8
5 -32 0
order = 2 x 3, 2 = no. of rows, 3 = no. of columns
Special Type of Matrices
1. Square Matrix
A m x n matrix is called as a square matrix if m = n i.e, no of rows = no. of columns
The elements
ij
a when i = j
? ?
11 22
a a ......... are called diagonal elements
Example:
??
??
??
12
45
2. Diagonal Matrix
A square matrix in which all non-diagonal elements are zero and diagonal elements may or
may not be zero.
Example:
??
??
??
10
05
Properties
a. diag [x, y, z] + diag [p, q, r] = diag [x + p, y + q, z + r]
b. diag [x, y, z] × diag [p, q, r] = diag [xp, yq, zr]
c.
? ?
?
??
?? ?
?? ??
??
1
1 1 1
diag x, y, z diag , ,
x y z
d.
? ?
??
??
t
diag x, y, z = diag [x, y, z]
e.
? ?
?? ?? ?
?? ??
n
n n n
diag x, y, z diag x ,y ,z
f. Eigen value of diag [x, y, z] = x, y & z
g. Determinant of diag [x, y, z] = xyz
3. Scalar Matrix
A diagonal matrix in which all diagonal elements are equal.
4. Identity Matrix
A diagonal matrix whose all diagonal elements are 1. Denoted by I
Properties
a. AI = IA = A
b.
?
n
II
c.
?
?
1
II
d. det(I) = 1
5. Null matrix
An m x n matrix whose all elements are zero. Denoted by O.
Properties:
a. A + O = O + A = A
b. A + (- A) = O
6. Upper Triangular Matrix
A square matrix whose lower off diagonal elements are zero.
Example:
??
??
??
??
??
3 4 5
0 6 7
0 0 9
7. Lower Triangular Matrix
A square matrix whose upper off diagonal elements are zero.
Example:
??
??
??
??
??
3 0 0
4 6 0
5 7 9
8. Idempotent Matrix
A matrix is called Idempotent if ?
2
AA
Example:
??
??
??
10
01
9. Involutary Matrix
A matrix is called Involutary if ?
2
AI .
Matrix Equality
Two matrices ? ?
? mn
A and ? ?
? p q
B are equal if
m = p ; n = q i.e., both have same size
ij
a =
ij
b for all values of i & j.
Addition of Matrices
For addition to be performed, the size of both matrices should be same.
If [C] = [A] + [B]
Then ??
ij ij ij
c a b
i.e., elements in same position in the two matrices are added.
Subtraction of Matrices
[C] = [A] – [B]
= [A] + [–B]
Difference is obtained by subtraction of all elements of B from elements of A.
Hence here also, same size matrices should be there.
Scalar Multiplication
The product of any m × n matrix A
??
??
??
jk
a and any scalar c, written as cA, is the m × n
matrix cA =
??
??
??
jk
ca obtained by multiplying each entry in A by c.
Multiplication of two matrices
Let ? ?
? m n
A and ? ?
? p q
B be two matrices and C = AB, then for multiplication, [n = p]
should hold. Then,
?
?
?
n
jk
j1
ik ij
C a b
Properties
? If AB exists then BA does not necessarily exists.
Example: ? ?
? 3 4
A , ? ?
? 4 5
B , then AB exits but BA does not exists as 5 ? 3
So, matrix multiplication is not commutative.
? Matrix multiplication is not associative.
A(BC) ? (AB)C .
? Matrix Multiplication is distributive with respect to matrix addition
A(B + C) = AB +AC
? If AB = AC ? B = C (if A is non-singular)
BA = CA ? B = C (if A is non-singular)
Transpose of a matrix
If we interchange the rows by columns of a matrix and vice versa we obtain transpose of a
matrix.
eg., A =
??
??
??
??
??
13
24
65
;
??
?
??
??
T
126
A
3 4 5
Conjugate of a matrix
The matrix obtained by replacing each element of matrix by its complex conjugate.
Properties
a.
? ?
? AA
b.
? ?
? ? ? A B A B
c.
? ?
? KA K A
d.
? ?
? AB AB
Transposed conjugate of a matrix
The transpose of conjugate of a matrix is called transposed conjugate. It is represented by
?
A .
a.
? ?
?
?
? AA
b. ? ?
?
??
? ? ? A B A B
c. ? ?
?
?
? KA KA
d. ? ?
?
??
? AB B A
Page 5
LINEAR ALGEBRA
MATRICES
A matrix is a rectangular array of numbers (or functions) enclosed in brackets. These numbers
(or function) are called entries of elements of the matrix.
Example:
??
??
??
2 0.4 8
5 -32 0
order = 2 x 3, 2 = no. of rows, 3 = no. of columns
Special Type of Matrices
1. Square Matrix
A m x n matrix is called as a square matrix if m = n i.e, no of rows = no. of columns
The elements
ij
a when i = j
? ?
11 22
a a ......... are called diagonal elements
Example:
??
??
??
12
45
2. Diagonal Matrix
A square matrix in which all non-diagonal elements are zero and diagonal elements may or
may not be zero.
Example:
??
??
??
10
05
Properties
a. diag [x, y, z] + diag [p, q, r] = diag [x + p, y + q, z + r]
b. diag [x, y, z] × diag [p, q, r] = diag [xp, yq, zr]
c.
? ?
?
??
?? ?
?? ??
??
1
1 1 1
diag x, y, z diag , ,
x y z
d.
? ?
??
??
t
diag x, y, z = diag [x, y, z]
e.
? ?
?? ?? ?
?? ??
n
n n n
diag x, y, z diag x ,y ,z
f. Eigen value of diag [x, y, z] = x, y & z
g. Determinant of diag [x, y, z] = xyz
3. Scalar Matrix
A diagonal matrix in which all diagonal elements are equal.
4. Identity Matrix
A diagonal matrix whose all diagonal elements are 1. Denoted by I
Properties
a. AI = IA = A
b.
?
n
II
c.
?
?
1
II
d. det(I) = 1
5. Null matrix
An m x n matrix whose all elements are zero. Denoted by O.
Properties:
a. A + O = O + A = A
b. A + (- A) = O
6. Upper Triangular Matrix
A square matrix whose lower off diagonal elements are zero.
Example:
??
??
??
??
??
3 4 5
0 6 7
0 0 9
7. Lower Triangular Matrix
A square matrix whose upper off diagonal elements are zero.
Example:
??
??
??
??
??
3 0 0
4 6 0
5 7 9
8. Idempotent Matrix
A matrix is called Idempotent if ?
2
AA
Example:
??
??
??
10
01
9. Involutary Matrix
A matrix is called Involutary if ?
2
AI .
Matrix Equality
Two matrices ? ?
? mn
A and ? ?
? p q
B are equal if
m = p ; n = q i.e., both have same size
ij
a =
ij
b for all values of i & j.
Addition of Matrices
For addition to be performed, the size of both matrices should be same.
If [C] = [A] + [B]
Then ??
ij ij ij
c a b
i.e., elements in same position in the two matrices are added.
Subtraction of Matrices
[C] = [A] – [B]
= [A] + [–B]
Difference is obtained by subtraction of all elements of B from elements of A.
Hence here also, same size matrices should be there.
Scalar Multiplication
The product of any m × n matrix A
??
??
??
jk
a and any scalar c, written as cA, is the m × n
matrix cA =
??
??
??
jk
ca obtained by multiplying each entry in A by c.
Multiplication of two matrices
Let ? ?
? m n
A and ? ?
? p q
B be two matrices and C = AB, then for multiplication, [n = p]
should hold. Then,
?
?
?
n
jk
j1
ik ij
C a b
Properties
? If AB exists then BA does not necessarily exists.
Example: ? ?
? 3 4
A , ? ?
? 4 5
B , then AB exits but BA does not exists as 5 ? 3
So, matrix multiplication is not commutative.
? Matrix multiplication is not associative.
A(BC) ? (AB)C .
? Matrix Multiplication is distributive with respect to matrix addition
A(B + C) = AB +AC
? If AB = AC ? B = C (if A is non-singular)
BA = CA ? B = C (if A is non-singular)
Transpose of a matrix
If we interchange the rows by columns of a matrix and vice versa we obtain transpose of a
matrix.
eg., A =
??
??
??
??
??
13
24
65
;
??
?
??
??
T
126
A
3 4 5
Conjugate of a matrix
The matrix obtained by replacing each element of matrix by its complex conjugate.
Properties
a.
? ?
? AA
b.
? ?
? ? ? A B A B
c.
? ?
? KA K A
d.
? ?
? AB AB
Transposed conjugate of a matrix
The transpose of conjugate of a matrix is called transposed conjugate. It is represented by
?
A .
a.
? ?
?
?
? AA
b. ? ?
?
??
? ? ? A B A B
c. ? ?
?
?
? KA KA
d. ? ?
?
??
? AB B A
Trace of matrix
Trace of a matrix is sum of all diagonal elements of the matrix.
Classification of real Matrix
a. Symmetric Matrix : ? ?
T
AA ?
b. Skew symmetric matrix : ? ?
T
AA ??
c. Orthogonal Matrix :
? ?
T 1 T
A A ; AA =I
?
?
Note:
a. If A & B are symmetric, then (A + B) & (A – B) are also symmetric
b. For any matrix
T
AA is always symmetric.
c. For any matrix,
??
??
??
??
T
A + A
2
is symmetric &
??
?
??
??
??
T
AA
2
is skew symmetric.
d. For orthogonal matrices, A1 ??
Classification of complex Matrices
a. Hermitian matrix :
? ?
AA
?
?
b. Skew – Hermitian matrix : AA
?
??
c. Unitary Matrix :
? ?
1
A A ;AA 1
? ? ?
??
Determinants
Determinants are only defined for square matrices.
For a 2 × 2 matrix
11 12
21 22
aa
aa
?? =
11 22 12 21
a a a a ?
Minors & co-factor
If
11 12 13
21 22 23
31 32 33
aaa
aaa
aaa
??
Read More
FAQs on Short Notes: Linear Algebra - Short Notes for Electrical Engineering - Electrical Engineering (EE)
| 1. What is linear algebra? |  |
Linear algebra is a branch of mathematics that deals with the study of vectors, vector spaces, linear transformations, and systems of linear equations. It focuses on studying the properties and operations of these mathematical structures and their applications in various fields such as physics, engineering, computer science, and data analysis.
| 2. What are the applications of linear algebra? | |
Linear algebra has a wide range of applications in various fields. Some of the common applications include:
- Physics: Linear algebra is used to describe physical systems and phenomena, such as quantum mechanics, electromagnetism, and fluid dynamics.
- Engineering: It is used in solving problems related to circuits, control systems, structural analysis, and optimization.
- Computer Science: Linear algebra is essential for computer graphics, image processing, machine learning, data mining, and cryptography.
- Economics: It is used in economic modeling, input-output analysis, and optimization problems.
- Statistics: Linear algebra is used in multivariate analysis, regression analysis, and principal component analysis.
| 3. What are vectors and vector spaces in linear algebra? | |
In linear algebra, a vector is a mathematical object that represents both magnitude and direction. It is typically represented as an ordered list of numbers. Vectors are used to represent physical quantities such as force, velocity, and displacement.
A vector space is a collection of vectors that satisfies certain properties. These properties include closure under addition and scalar multiplication, existence of zero vector and additive inverse, and associativity and distributivity properties. Vector spaces provide a framework for performing operations on vectors and studying their properties.
| 4. What are linear transformations in linear algebra? | |
A linear transformation is a function between two vector spaces that preserves vector addition and scalar multiplication. In other words, it maps vectors from one vector space to another while preserving their linear properties.
Linear transformations can be represented by matrices. The matrix representation of a linear transformation allows us to perform operations on vectors using matrix multiplication. Linear transformations have applications in areas such as computer graphics, image processing, and data analysis.
| 5. How is linear algebra used in solving systems of linear equations? | |
Linear algebra provides powerful tools for solving systems of linear equations. By representing the system of equations as a matrix equation, we can use matrix operations to solve for the unknown variables.
The process involves transforming the system of equations into an augmented matrix, performing row operations to bring it into row-echelon or reduced row-echelon form, and then back-substituting to find the values of the unknown variables.
This method is widely used in various fields, such as engineering, physics, and economics, for solving problems that involve multiple linear equations with multiple unknowns.
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