3A.7 System of Linear Equations

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Consider the following set of linear algebraic equations:

a11x1 þ a12x2 þ · · · þ a1nxn ¼ c1

a21x1 þ a22x2 þ · · · þ a2nxn ¼ c2

.. .

am1x1 þ am2x2 þ · · · þ amnxn ¼ cm

We need to solve for x1; x2; …; xn:

This problem can be expressed in the vector– matrix form:

Am£nxn ¼ cm B ¼ ½A; c􀀉

A solution exists iff rank½A; c􀀉 ¼ rank½A􀀉

Two cases can be considered:

Case 1:

If m $ n and rank½A􀀉 ¼ n ) unique solution for x:

Case 2:

If m # n and rank½A􀀉 ¼ m ) infinite number of solutions for x

x ¼ AHðAAHÞ21C ( minimum norm form

Specifically, out of the infinite possibilities, this is the solution that minimizes the norm, xHx:

Note that the superscript H denotes the Hermitian transpose, which is the transpose of the complex

conjugate of the matrix.

For example,

A ¼

1 þ j 2 þ 3j 6

3 2 j 5 21 2 2j

" #

Then

AH ¼

1 2 j 3 þ j

2 2 3j 5

6 21 þ 2j

2

664

3

775

If the matrix is real, its Hermitian transpose is simply the ordinary transpose.

In general, if rank½A􀀉 # n ) infinite number of solutions.

The space formed by solutions Ax ¼ 0 ) is called the null space

dimðnull spaceÞ ¼ n 2 k where rank½A􀀉 ¼ k

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