HESC686 Mathematics and Signal Processing for Biomechanics
Introduction to matrices
Matrix = 2-D array.
Matrices are, among other things, shorthand representations of systems of linear equations.
We can use matrix notation to represent and solve the system of linear equations from the previous
lecture:
x–y+z=0
2x - 3y + 4z = -2
-2x – y + z = 7
This can be written with matrices as follows:
1 −1 1 𝑥
0
[ 2 −3 4] [𝑦] = [−2]
−2 −1 1 𝑧
7
once we know the rules for multiplying a matrix times a column vector to get another column
vector. The matrix equation above can be written
𝑨𝒙 = 𝒃
where
𝑥
1 −1 1
0
𝑨 = [ 2 −3 4] , 𝒙 = [𝑦] , 𝒃 = [−2]
𝑧
−2 −1 1
7
We can solve this system of 3 equations and 3 unknowns using Matlab. First define the constants
A (a matrix) and b (a column vector) in Matlab:
>> A=[1 -1 1; 2 -3 4; -2 -1 1]
>> b=[0; -2; 7]
Then solve the system by defining an augmented matrix (Aaug) and using Matlab’s “rowreduction echelon formula” function (rref) to find x:
>> Aaug=[A b]
>> x=rref(Aaug)
At this point, the above method may seem mysterious. In the next lecture we will see another way
to solve the system (i.e. another way to find x) which will give more insight and make more sense,
but first we must learn more about working with matrices.
Matrix basics
Dimensions are given as rows (first), columns (second).
m rows, n columns is “m by n”
Square matrix has equal number of rows and columns.
One-by-n matrix = row vector; n-by-one matrix = column vector.
Matrix elements
Notation: aij = element in row i, column j.
Diagonal elements = aii.
Equality of matrices
Two matrices are equal if they have same dimensions and all elements are the same.
Matrix arithmetic
Matrix addition
Add corresponding elements; matrix dimensions must match.
Commutative: A+B=B+A
Associative: (A+B)+C = A+(B+C)
Zero matrix: all elements = 0. A + 0 = 0 + A = A
Matrix subtraction: like addition.
Multiplication (or division) by a scalar
Multiply (or divide) each element by same scalar. Note cA = Ac .
Matrix multiplication
Am×n Bn×ℓ = Cm× ℓ where the subscript “m×n” indicates that matrix A has m rows
and n columns, etc.
The element cij, in row i, column j of the resulting matrix C, is given by
cij = ai1b1j + ai2b2j + … + ainbnj = ∑
𝑛
𝑘=1
aik bkj
1 2
5 6
),B = (
) , C = AB =?
3 4
7 8
Note that number of columns in A must equal the number of rows in B for matrix
multiplication (AB) to work.
1 0
Identity matrix: 𝐼2 = (
). AI = IA = A .
0 1
Matrix multiplication is not commutative. Use Matlab to compare D=BA to
C=AB.
>>A=[1 2;3 4];
>>B=[5 6;7 8];
>>C=A*B, D=B*A
Matrix division
This is complicated. We will return to this topic later.
Example: 𝐴 = (
We can use a matrix and vector to represent a linear system with n equations and n unknowns.
(See also the example at the start of this lecture.) In this example, A is an n-by-n matrix and x and
b are coulmn vectors of length n:
𝑨𝒙 = 𝒃
𝑥
𝑦
where A is an n-by-n matrix and x and b are coulmn vectors of length n, with 𝐱 = ( ) .
𝑧
Copyright © 2010 William C. Rose