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Matrix Algebra

ENGIN 211, Engineering Math

Matrix in Circuit Analysis

Example: Mesh Analysis

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Apply the Kirchhoff voltage law:

Reorganize:

Use matrix: Solution:

Matrix

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Singular and Nonsingular Matrices

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(3)-(4),

If then is singular, otherwise nonsingular.

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.

Matrix Determinant Example

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Method 1

Method 2

Method 3

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.

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

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.

Inverse Matrix

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Unit matrix If then

Consider this set of equations:

or

Inverse Method (1)

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Note: it requires matrix determinant be nonzero.

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.

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.

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

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.

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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