EG1C2 Engineering Maths : Matrix Algebra Dr Richard Mitchell, Department of Cybernetics

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EG1C2 Engineering Maths Matrix AlgebraDr
Richard Mitchell, Department of Cybernetics

• Aim Describe matrices and their use in varied
  applications
• Syllabus
• Introduction why use definitions simple
  processing
• Determinants and inverses
• Two Port Networks, for electronics other
  systems
• Gaussian elimination to solve linear equations
  Gauss-Jordan
• Matrix Rank and Cramer's Rule and Theorem
• Eigenvalues and eigenvectors, applications incl.
  state space
• Vectors - and their relationship with matrices.
• References
• K.A.Stroud Engineering Mathematics Fifth Edn
  - Palgrave
• Glyn James - Modern Engineering Mathematics -
  Addison Wesley
• Online Notes http//www.cyber.rdg.ac.uk/people/R.M
  itchell/teach.htm

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Matrix Algebra - Introduction

• Simple systems - defined by equations y f(x)
  e.g. y kx
• Many systems involve many variables
• y1 k11x1 k12x2 k13x3
• y2 k21x1 k21x2 k23x3
• y3 k31x1 k32x2 k33x3 etc.
• Matrix techniques allow us to represent these by
  y k x
• Bold letters show these are vector or matrix
  quantities.
• Why we use matrices
• Can group related data and process them together.
• Can use clever techniques to solve problems.
• Standard matrix manipulation techniques are
  available.
• Can use a computer to process the data e.g. use
  MATLAB?.
• In course use only 2 or 3 variables, use
  computers for more.

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Example Suspended Mass
T1 T2 are tensions in two wires. (Angles chosen
for easy arithmetic)

• Resolving forces in horizontal and vertical
  directions
• cos (16.26) T1 cos (36.87) T2
• sin (16.26) T1 sin (36.87) T2 300 weight
  of mass
• Simplifies to 0.96 T1 - 0.8 T2 0 and 0.28
  T1 0.6 T2 300

The system can then be written as A.T Y T, Y
are vectors - 1 column 2 rows, A is a matrix - 2
columns 2 rows
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Example Electronic Circuit

• Using Kirchhoffs Voltage and Current Laws
• First Loop 12 18 i1 10 i2 or 18 i1
  10 i2 12
• Second Loop 10 i2 15 i3 or -10 i2
  15 i3 0
• Summing currents i1 i2 i3 or -i1
  i2 i3 0

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Matrix Definitions

• Rectangular array of numbers, complex numbers,
  functions, ..
• If r rows c columns, rc elements, rc is order
  of matrix.

A is n m matrix Square if n m aij is in row i
column j
A vector has one column or one row
Square matrix a11, a22, .. ann form the main
diagonal
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Simple Matrix Operations
Illustrate these by defining A (size mn) and B
(size rc)

• Equality A and B are identical if,
• they are of the same size, m r and n c, and
• corresponding elements are same ie aij bij
  for all i,j

A B, but A ? C, A ? D
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Matrix Addition
A and B must have same size result is a matrix
also of the same size, call it matrix R, in which
for all elements, rij aij bij.
NB A B B A (A B) C A (B C)
A B C
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Matrix Subtraction

• A and B must have same size result is a matrix
  also of the same size, call it R, in which for
  all elements, rij aij - bij.

Matrix Scalar Multiplication
Each element in the matrix is multiplied by a
scalar constant R k.A Thus, each rijk.aij.
Note, k (A B) kA kB
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Matrix Multiplication

• R AB, number of columns in A number of rows
  in B
• R has number of rows as A and number of columns
  as B.
• e.g if A is 2 3, B is 3 4 , then A B is 2
  4 matrix.

• Do first element of ith row of A first element
  of jth column of B
• Multiply second, third, etc. elements of these
  rows and columns
• Find the sum of each product and store in rij
• If A B ok, then B A is only possible if A B
  are square.
• A B ? B A in general. A(BC) (AB)C
  ABC
• A(BC) (AB)(AC) (kA)Bk(AB)A(kB)
  scalar k

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Examples Exercise
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Multiplication and Example Systems
Suspended Mass A 22 matrix times a 21 vector
a 21 vector
Electronic Circuit A 33 matrix times a 31
vector a 31 vector
Exercise Express equations 5x 2y 16 and
3x 18 4y in matrix form
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Matrix Transpose (A transposed is AT)
If R AT, then rij aji. If A is size mn,
then AT is size nm.
Note (AT)TA (AB)TATBT (AB)TBTAT
(kA)TkAT If ATA, A is symmetrix matrix. If
AT-A, A is skew-symmetrix matrix