Matrix%20Algebra%20-%20Introduction%20Continued

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Matrix Algebra - Introduction Continued
Special Matrices
If S scalar, A S S A. A I A To
convert a scalar, k, to a matrix, multiply scalar
by I
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Comments on Diagonal/Triangular Matrices
It is easy to evaluate - clearly x 4, y 5 and
z 9
It is quite easy to evaluate Clearly z 2 from
the 3rd row. Then, row 2 gives 2y 42 18
But z known, so y 5. Then, row 1 gives x
35 22 21 x 2
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What if matrix not triangular/diagonal?
It turns out that there is a rather useful
matrix, such that
So pre-multiply both sides of equation by the
arbitrary matrix
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Multiplying out the matrices we get a simplified
equation
This, of course, is an equation we solved
earlier. Thus the solution to the equation is x
2, y 5 and z 2. If pre-multiply one side of
eqn, must do same to other side.
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Application Stochastic Matrix Markov Process

• In 1995 30 of graduates become researchers (R),
  20 get jobs in commercial sector (C) and 50
  join industry (I).
• Over 5 years this changes according to the
  following table each element is the probability
  of transition.
• To R C I
• From R 0.7 0.1 0.2 (e.g. 0.2 prob of R to I)
• C 0.1 0.6 0.3 (e.g. 0.6 prob of stay C)
• I 0.1 0.1 0.8 (NB Rows add up to 1)

This can be put in matrix form, a so-called
Stochastic Matrix
Let vector for numbers doing jobs in 1995 be
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Then the job situation in year 2000 is found by
If the same transition matrix applies, the jobs
in 2005 are
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We can find situation back in 1990
Post-multiplying by another magic matrix
Hence 200 6R, so R 33.33 140
2.4R 3C so C (140-80)/3 20 100
R C I so I 46.67
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Remember, (AB)TBTAT By transposing the
matrices (note order), the equation becomes
This will be used later. Note, to find situation
in 1990, equation is
And we then pre-multiply to get
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Application 2D CAD package
Draw logic circuits - first define gates, then
have circuit with them

• AND gate size 100,100 bottom left corner at 0,0.
  
• On drawing, AND gate is size 2525 at 25,25 must
  transform
• 0,0 on gate 25,25 on drawing 0,100 ? 25,50
  50,0 ? 37.5, 25

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This is achieved by scaling and translating each
point scaling x x / 4 y y /
4 translating x x 25 y y
25 overall x x / 4 25 y y / 4
25 In general want x x Sx y y
Sy x x Dx y y Dy In matrix form,
point x,y defined by x y 1 1 is dummy
element so have square matrices for
multiplication.
Scaling matrix - to scale x by Sx and y by Sy
Translation matrix - to translate in x by Dx and
in y by Dy
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To draw AND gate, transformation matrix
scalingtranslation
Then any point x,y on the AND gate is transformed
to x',y', by
For drawing the NOT gate, we need a rotation
matrix also
Rotation matrix - by angle A anticlockwise
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To draw the NOT gate scale by 0.25,0.25,
rotate by 90o, translate by 100,50 thus
Then any point x,y on the NOT gate is transformed
to x',y', by
Exercise To draw NAND gate, half size, rotated
by 180O at 50,100