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Matrix Decomposition

Matrix Decomposition

Author: c o
Description:

To introduce triangular matrices and LU-Decomposition
To learn how to use an algorithmic technique in order to decompose arbitrary matrices
To apply LU-Decomposition in the solving of linear systems

This packet introduces triangular matrices, and the technique of decomposing matrices into triangular matrices in order to more easily solve linear systems.

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Tutorial

Triangular Matrices

Before beginning with this packet, you should be comfortable with matrix multiplication, Gaussian elimination, the definition of the determinant of a matrix (see also here), and solving linear systems.

Definition

 

The Determinant Of Triangular Matrices

The determinant of a triangular matrix, either upper or lower, and of any size, is just the product of its diagonal entries.  This single property immensely simplifies the ordinarily laborious calculation of determinants.  Here are some examples for 2x2 and 3x3 matrices.

Lower Triangular 2x2 Matrix

 has a determinant of ad-c0 = ad.

Upper Triangular 3x3 Matrix

 which has a determinant of a(ej - 0f) - b(0j - 0f) + c(0 - 0e) = aej.

Here is a numerical example of the same thing.

  which has the determinant 1(2*2 - 0*1) - 2(0*2 - 0*1) + 3(0*0 - 0*2) = 1*2*2 = 4, which is just the product of the diagonal entries.

Decomposing Matrices

LU-Decomposition

Solving Linear Systems

Finding A Decomposition

In this example we find an LU Decomposition for a matrix.

Checking Our Work

Here is a screen capture using the free computer algebra package called maxima to check our work:

In the above, we define L2 and U2 from the video, then multiply them to get our initial matrix M, showing that M=L2U2 is an LU-decomposition.

Applying LU-Decomposition To Linear Systems

We use LU-Decomposition to solve a linear system.