MATRIX METHODS
SYSTEMS OF
LINEAR EQUATIONS
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MATRIX METHODS
SYSTEMS OF
LINEAR EQUATIONS
System of Linear Equations
- (^) Determining the values of
x 1 , x 2 , .....xn, that
simultaneously satisfy a
set of equations
Mathematical Background
Matrix Notation
- (^) a horizontal set of elements is called a row
- (^) a vertical set is called a column
- (^) first subscript refers to the row number
- (^) second subscript refers to column number
A
a a a a a a a a
a a a a
n n
m m m mn
11 12 13 1 21 22 23 2
1 2 3
m m m mn
n
n
a a a a
a a a a
a a a a
A
...
....
...
...
1 2 3
21 22 23 2
11 12 13 1
This matrix has m rows an n column.
It has the dimensions m by n (m x n)
note subscript
Row vector: m=
Column vector: n=1 Square matrix: m = n
B ^ ^ b 1^ b 2^ ....... bn
C
c c
cm
1 2 . .
A
a a a a a a a a a
11 12 13 21 22 23 31 32 33
A
a a a a a a a a a
11 12 13 21 22 23 31 32 33
The diagonal consist of the elements
a 11 a 22 a 33
- (^) Symmetric matrix
- (^) Diagonal matrix
- (^) Identity matrix
- (^) Upper triangular matrix
- (^) Lower triangular matrix
- (^) Banded matrix
Diagonal Matrix
A square matrix where all elements off
the main diagonal are zero
A
a a a a
11 22 33 44
Identity Matrix
A diagonal matrix where all elements on
the main diagonal are equal to 1
A ^
The symbol [I] is used to denote the identify matrix.
Lower Triangular Matrix
All elements above the main diagonal are
zero
A ^
Banded Matrix
All elements are zero with the exception
of a band centered on the main diagonal
A
a a a a a a a a a a
11 12 21 22 23 32 33 34 43 44
Matrix Operating Rules
- (^) Multiplication of a matrix [A] by a scalar g
is obtained by multiplying every element of [A] by g
B ^ g ^ A
ga ga ga ga ga ga
ga ga ga
n n
m m mn
11 12 1 21 22 2
1 2
Matrix Operating Rules
- (^) The product of two matrices is represented as
[C] = [A][B]
n = column dimensions of [A] n = row dimensions of [B]
c (^) ij a (^) ik bkj k
N
1
Matrix multiplication
- (^) If the dimensions are suitable, matrix
multiplication is associative ([A][B])[C] = A
- (^) If the dimensions are suitable, matrix
multiplication is distributive ([A] + [B])[C] = [A][C] + [B][C]
- (^) Multiplication is generally not commutative
[A][B] is not equal to [B][A]
Inverse of [A]
A A A A I
1 1