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Matrix Algebra
ENGIN 211, Engineering Math
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Matrix in Circuit Analysis
Example: Mesh Analysis
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Apply the Kirchhoff voltage law:
Reorganize:
Use matrix: Solution:
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Matrix
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Singular and Nonsingular Matrices
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(3)-(4),
If then is singular, otherwise nonsingular.
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Matrix Determinant
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In general
where is the determinant of a matrix obtained
by eliminating the i-th row and j-th column.
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Matrix Determinant Example
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Method 1
Method 2
Method 3
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Rank of Matrix and Equivalent Matrices
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The following elementary row operations on matrix A produces a new
row equivalent matrix B which has the same order and rank as those of
matrix A:
1) Interchange of two rows
2) Multiply each element of a row by the same non-zero scalar
3) Adding or subtracting corresponding elements of two rows
There are also three similar elementary column operations to form
column equivalent matrix.
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Solutions of Equations
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Consistency test for n equations with n unknowns
1) A unique set of solutions if rank A = rank Ab =n
2) Infinite number of solutions if rank A = rank Ab < n
3) No solutions if rank A < rank Ab
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Examples
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No solutions
In fact, Eq(1) and Eq(3) conflict.
Case 1:
Case 2:
Infinite solutions
In fact, Eq(1) and Eq(3) are the same equation.
Case 3:
A unique set of solutions.
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Inverse Matrix
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Unit matrix If then
Consider this set of equations:
or
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Inverse Method (1)
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Note: it requires matrix determinant be nonzero.
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Inverse Method 2 - Row Transformation
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Gaussian Elimination Method
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Eigenvalues and Eigenvectors
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Eigenvalues and Eigenvectors
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Example
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Matrix Diagonalization
17 Why is it useful? It is used often in coordinate system transformations.
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What is it for?
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1) Matrix diagonalization is the process of taking a square matrix and
converting it into a diagonal matrix - that shares the same fundamental
properties of the underlying matrix.
2) Matrix diagonalization is equivalent to transforming the underlying
system of equations into a special set of coordinate axes in which the
matrix takes this diagonal form.
3) Diagonalizing a matrix is also equivalent to finding the
matrix's eigenvalues, which turn out to be precisely the entries of the
diagonalized matrix. Similarly, the eigenvectors make up the new set of
axes corresponding to the diagonal matrix.
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Coordinate System Transformation
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P(x,y) Q(u,v)
y
x
v
u
If every point in (x,y) system corresponds to a point in (u,v) system
with a simple scaling relationship: 𝑢 = 𝑎𝑥, 𝑣 = 𝑏𝑦 , then the
following matrix transformations allows for going back and forth
between the two coordinate systems,
𝑢𝑣
=𝑎 00 𝑏
𝑥𝑦 ,
𝑥𝑦 =
1/𝑎 00 1/𝑏
𝑢𝑣
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Rotation of Axes
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θ
θ
y
x
u
v P
𝑥𝑦 =
cos𝜃 −sin𝜃sin𝜃 cos𝜃
𝑢𝑣
𝑢𝑣
=cos𝜃 sin𝜃−sin𝜃 cos𝜃
𝑥𝑦
The transformations are as follows:
The same point P can be described
in two systems (x,y) and (u,v) that
are rotated by θ.
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Summary Key points:
Matrix singularity
Matrix determinant
Rank of matrix and equivalent matrices
Matrix used for solutions of equations
Inverse matrix
Gaussian elimination
Supplemental: • Eigenvalues and eigenvectors
• Matrix diagnolization
• Matrix transformation
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