Sign in

username:

password:



Not a member?

Search Online Books



Search tips

Free Online Books



Chapters

See Also

Embedded SystemsFPGAElectronics

Mathematics of the DFT
    Matrices
       Solving Linear Equations Using Matrices

Chapter Contents:

Search Mathematics of the DFT

  

Book Index | Global Index

Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?

  

Solving Linear Equations Using Matrices

Consider the linear system of equations

\begin{eqnarray*} a x_1 + b x_2 &=& c \\ d x_1 + e x_2 &=& f \end{eqnarray*}

in matrix form:

$\displaystyle \left[\begin{array}{cc} a & b \\ [2pt] d & e \end{array}\right] \... ..._2 \end{array}\right] = \left[\begin{array}{c} c \\ [2pt] f \end{array}\right] $

This can be written in higher level form as

$\displaystyle \mathbf{A}\underline{x}= \underline{b}, $

where $ \mathbf{A}$ denotes the two-by-two matrix above, and $ \underline{x}$ and $ \underline{b}$ denote the two-by-one vectors. The solution to this equation is then

$\displaystyle \underline{x}= \mathbf{A}^{-1}\underline{b}= \left[\begin{array}{... ...\end{array}\right]^{-1}\left[\begin{array}{c} c \\ [2pt] f \end{array}\right]. $

The general two-by-two matrix inverse is given by

$\displaystyle \left[\begin{array}{cc} a & b \\ [2pt] d & e \end{array}\right]^{... ...rac{1}{ae-bd}\left[\begin{array}{cc} e & -b \\ [2pt] -d & a \end{array}\right] $

and the inverse exists whenever $ ae-bd$ (which is called the determinant of the matrix $ \mathbf{A}$) is nonzero. For larger matrices, numerical algorithms are used to invert matrices, such as used by Matlab based on LINPACK [23]. An initial introduction to matrices and linear algebra can be found in [45].

Order a Hardcopy of Mathematics of the DFT

Previous: Matrix Multiplication
Next: Matlab/Octave Examples
written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


Comments

No comments yet for this page


Add a Comment
You need to login before you can post a comment (best way to prevent spam). ( Not a member? )