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Matrix Algebra
Matrix Algebra
• Definitions
• Addition and Subtraction
• Multiplication
• Determinant
• Inverse
• System of Linear Equations
• Quadratic Forms
• Partitioning
• Differentiation and Integration
2.1 Definitions
• Scalar (0D matrix)
• Vector ( 1D matrix )
Transpose
• Matrix ( 2D matrix )
Transpose
2.1 Definitions, contd
• Matrix size, [m x n] or [i x j], ( [row x col] )
– Vector, [m x 1]
[m x n] = [4 x 2]
• Square matrix, (m=n) [m x m]
• Symmetric matrix, B=BT
2.1 Definitions, contd
• Diagonal matrix
• Identity matrix, (unit matrix),
• Zero matrix
AI=A
2.2 Addition and Subtraction
• Vector
• Matrix
2.3 Multiplication - scalar
• Scalar – vector multiplication
• Scalar – matrix multiplication
2.3 Multiplication - vector
• Scalar product, (vector – vector multiplication)
[1 x n] [n x 1] = [1 x 1]
• Length of vector (cf. Pythagoras' theorem)
2.3 Multiplication - vector
• Matrix product
 a1 
ab T = a2 [b1 b2
 a3 
 a1b1
b3 ] = a2b1
 a3b1
[m x 1] [1 x n] =
a1b2
a2b2
a3b2
[m x n]
a1b3 
a2b3 
a3b3 
2.3 Multiplication - matrix
• Matrix – vector multiplication
[m x n]
[3 x 2]
[n x 1] =
[2 x 1] =
[m x 1]
[3 x 1]
• Vector – matrix multiplication
[1 x m] [m x n] = [1 x n]
2.3 Multiplication - matrix
• Matrix – matrix multiplication
[m x n] [n x p] = [m x p]
Note!
2.3 Multiplication – matrix, contd
• Product of transposed matrices
• Distribution law
2.4 Determinant
• The determinant may be calculated for any square
matrix, [n x n]
• Cofactor of matrix, A (i=row, k=column)
• Expansion formula
2.4 Determinant, examples
Cofactors
1 2
A=

3
4


c
= (−1) (1+1) 4 = 4
A11
c
A12
= (−1) (1+ 2 ) 2 = −2
Determinant
det A = 1 ⋅ 4 + 2(−4) = −2
1 2 3 
B = 4 5 6
7 8 5
Cofactors
c
= (−1) (1+1) (5 ⋅ 5 − 6 ⋅ 8) = −23
B11
B c21 = (−1) ( 2+1) (2 ⋅ 5 − 3 ⋅ 8) = −14
c
= (−1) (3+1) (2 ⋅ 6 − 3 ⋅ 5) = −3
B 31
Determinant
det B = 1(−23) + 4 ⋅14 + 7(−3) = 12
2.5 Inverse Matrix
• The inverse A-1 of a square matrix A is defined by
• The inverse may be determined by the cofactors, where
is the adjoint of A and
2.4 Inverse, examples
1 2
A=

3
4


 4 − 2
adjA = 

−
3
1


 4 − 2 1
A = adjA / det A = 
 (−2)
−
3
1


−1
1 2 3 
B = 4 5 6
7 8 5
What happens if
det A = 0 ?
− 23 22 − 3
adjB =  − 14 − 16 6 
 − 3
6 − 3
 − 23 14 − 3
1


−1
B = adjB / det B = − 22 − 16 6 
12
 − 3
6 − 3
T
2.6 Systems of Linear Equations
-number of equations is equal to number of unknowns
• Linear equation system
A is a square matrix [n x n],
x and b [n x 1] vector
b = 0 => Homogeneous system
b ≠ 0 => Inhomogeneous system
• Assume that det( A ) ≠ 0
2.6 Linear Equations
-number of equations is equal to number of unknowns
• Homogeneous system b = 0, (trivial solution: x = 0)
Eigenvalue problems
• Inhomogeneous system b ≠ 0
2.8 Quadratic forms and positive
definiteness
• If
• then the matrix A is positive definite
• If A is positive definite, all diagonal elements must
be positive
2.9 Partitioning
The matrix A may be partitioned as
and if
A may be written
2.9 Partitioning, contd
An equation system can be written
with
introduce
The partitioned equation system is written
or
2.10 Differentiation and integration
• A matrix A
• Differentiation
• Integration