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Basic Linear Algebra
Objectives:
- Write a matrix and know it’s size
- Add matrices
- Subtract matrices
- Multiply matrices
- Define a matrix transpose
- Define an Identity matrix
- Define a symmetric matrix
Matrix methods are a subset of linear algebra.
A matrix has a size defined by its number of rows and columns.
A=
The size of matrix A is 3 rows by 2 columns.
In common language it is said, "matrix A is 3 by 2".
Any term in the matrix is referred to by its row and column position. The row position is always given first. (i.e. A (2,2) = 1.0)
The diagonal of a matrix is defined as all terms where the row number and the column number are the same. (i.e. A (1,1) and A (2,2)).
Matrices can be added. Matrices MUST be the same size.
The resulting values is the sum of the corresponding terms
B=
Then: A +B=C Gives the result:
C=
Subtraction is handled in a similar way.
D =A − B gives the result
D=
Matrices can be multiplied, but not divided.
Matrix multiplication is NOT commutative. (it is order dependent).
The terms of the result consist of multiplying a row of the first matrix times a column of the second matrix.
G =A* F
F=
The 1,1 term (row one, column one) of G is:
G(1,1)=1.23.4+(2.3)7.3= 12.
The location where the result goes is dictated by the row and column number being multiplied.
The result G (3,2) comes from multiplying row 3 of A times column 2 of F. The final matrix G is:
G=
Matrix sizes must match:
Mustbe same.
(NxL) (NxM) (MxL)
G = A * F
Multiplication is associative and distributive.
A BC=(AB)C=A(B C) and (A+B)C=AC+B*C
A −^1 Ax=A −^1 b
I x=A −^1 b
x =A −^1 b
In practice the inverse is never calculated, a solution process is used.
There are two types of solution methods used in practice: 1) Direct and 2) Iterative. Direct solutions can be thought of as variations of the Gauss Elimination method. The iterative schemes can be thought of as variations of the Gauss-Sidel solution technique.
The transpose is defined as switching the rows with the columns. A superscript T designates the transpose. For example, the transpose of matrix H is H T :
H =
H = T
(3 by 4) (4 by 3)
The transpose can also be taken of a set of multiplied matrices by reversing the order of the multiplications and individually transposing the matrices.
( HAD)=DTAT HT
T
A matrix is symmetric when the rows are equal to the corresponding column. In other words, the matrix is equal to its transpose.
M=
