Matrix cookbook - Part 2

Today’s entry is continuation of my last post where I have introduced some basic information about matrices. In the next few lines I would like to share with you more definitions related to matrices and describe basic operations on it.

Let me start with definitions.

• Stochastic matrix is matrix whose sum of all row elements (or column elements) equal to 1.
• Triangularl matrix is matrix in which all elements above or below the principal have 0 value.
• Rank of matrix is equal to the number of linearly independent rows or columns it contains, whichever of these two numbers is smaller. It is worth to remember that rank of a square matrix is equal to the number of nonzero rows.
• Determinant is function that associates a scalar to a square matrix.
• Invertible matrix is matrix with a nonzero determinant.
• Regular matrix is a square matrix that determinant is not 0.
• Singular matrix is matrix that determinant is equal to 0.

Let me now show few basic operation on matrices. Let’s assume we have to matrices A and B.

• Addition
  
• Substraction
  
• Multiplication
  
• Calculation of matrix determinant
  

Matrix cookbook - Part 1

In this post I would like to introduce some basic information about matrices and matters relating to them. I do not pretend to make a comprehensive review out of it. Material I have prepared is divided into two parts and limited to what I think will be the most relevant to understand posts about Singular Value Decomposition, Latent Semantic Indexing and Vector Space Model I am going to write in the coming weeks. Goal of this post is just to help readers to grasp fundamental concepts and fully benefit from the future posts.

First of all, let’s go through some one-line definitions and visualizations that will introduce different types of matrices.

• Matrix is a rectangular array composed of rows and columns.
• Square matrix is matrix where the number of rows (m) equals to the number of columns (n).
• Principal of the matrix is diagonal extending from the upper-left to the lower-right corner of a square matrix.

• Trace of the matrix is sum of the all elements on the principal.
• Diagonal matrix is matrix in which all elements that are not located on the principal equal to 0.
• Scalar matrix is a variation of a diagonal matrix where all elements located on the principal are equal.
• Identity matrix also known as unit matrix is special case of scalar matrix where all elements are equal to 1.

• Transpose matrix is matrix that was obtained by converting rows into columns and vice versa.

Your suggestion for additional content or elaboration of some topics is most welcome. Please don’t hesitate to contact with me tomek@codequest.eu.

New Year’s resolution

I don’t usually make New Year’s Resolutions. I just feel if there is something I really want to do I will commit to it any time of the year. 

Few months ago I decided to arrange some space in the Internet where I will write about my professional interests and share my knowledge in the area of information retrieval and software engineering. There is a saying in the academic world suggesting that we teach in order to learn, that is, we don’t know what we don’t know until we try to teach it. I am aware of the fact I am still a newbie and there is lot of things I should learn and master. I hope this blog will help me to find the holes in the knowledge about things I think I have already got well in hand. I also would like to learn from all friends who will read this blog. 

It took some time before blog was designed and ready to write posts. In the meantime I went on a long trip to Thailand and Cambodia. I came back one week ago and just after Christmas I returned to the idea of blogging. I thought it will be fun to write first post on the January 1th, 2008.