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Null space and Rank are added

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File decompose-eigenvaluesrelated.tex

 
 
 
-\section{Takagi factorization}
 
-
-

File decompose-matrixproperties.tex

-\documentclass[a4paper,10pt]{extreport}
-\usepackage{amssymb,amsfonts,amsmath,cite,enumerate}
-\setcounter{tocdepth}{2} %%where n is the level,starting with 0 (chapters only)
-\usepackage{makeidx}
-\makeindex 
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-%%% this one adds the keyword 
-%command: an italic emphasize and addition to the index.
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-\geometry{inner=4.5cm} \geometry{outer=4cm}
-\geometry{top=0.6cm} \geometry{bottom=9cm}
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-\begin{document}
+% \documentclass[a4paper,10pt]{extreport}
+% \usepackage{amssymb,amsfonts,amsmath,cite,enumerate}
+% \setcounter{tocdepth}{2} %%where n is the level,starting with 0 (chapters only)
+% \usepackage{makeidx}
+% \makeindex 
+% \newcommand{\keyword}[1]{\textit{#1}\index{#1}} 
+% %%% this one adds the keyword 
+% %command: an italic emphasize and addition to the index.
+% \newcommand{\matlab}[1]{\texttt{#1}} %%%% this one adds the keyword 
+% 
+% 
+% \usepackage{geometry}
+% \geometry{inner=4.5cm} \geometry{outer=4cm}
+% \geometry{top=0.6cm} \geometry{bottom=9cm}
+% 
+% \begin{document}
 
 
 \chapter{Appendix: Basic Matrix Properties}
 
 
 
+\section{Rank of a matrix}
+The following are all equivalent statements\cite{hogbenhandbook} about a matrix $A\in \mathbb{F}^{m\times n}$.
+\begin{itemize}
+\item
+  The rank of $A$ is k.
+\item
+  dim(range($A$)) = k.
+\item
+  The reduced row echelon form of $A$ has k pivot columns.
+\item
+  A row echelon form of $A$ has k pivot columns.
+\item
+  The largest number of linearly independent columns of $A$ is k.
+\item
+  The largest number of linearly independent rows of $A$ is k.
+\end{itemize}
 
+The column rank of a matrix $A$ is the maximum number of linearly independent column vectors of $A$. The row rank of a matrix $A$ is the maximum number of linearly independent row vectors of $A$. Equivalently, the column rank of $A$ is the dimension of the column space of $A$, while the row rank of $A$ is the dimension of the row space of $A$.
 
+A result of fundamental importance in linear algebra is that the column rank and the row rank are always equal
 
+\subsection{Applications}
+One useful application of calculating the rank of a matrix is the computation of the number of solutions of a system of linear equations. According to the Rouche-Capelli theorem, the system is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix. If, on the other hand, ranks of these two matrices are equal, the system must have at least one solution. The solution is unique if and only if the rank equals the number of variables. Otherwise the general solution has k free parameters where k is the difference between the number of variables and the rank. In control theory, the rank of a matrix can be used to determine whether a linear system is controllable, or observable.
 
-\bibliographystyle{unsrt}  \bibliography{biblio/kmvlinalgdecompose}
-\printindex
-\end{document}
+\subsection{Computing the Rank of the Matrix in MATLAB}
+The rank function provides an estimate of the number of linearly independent rows or columns of a full matrix.
+
+k = rank(A) returns the number of singular values of A that are larger than the
+default tolerance, max(size(A))*eps(norm(A)).
+
+k = rank(A,tol) returns the number of singular values of A that are larger than \textit{tol}.
+
+\paragraph{Algorithm}
+There are a number of ways to compute the rank of a matrix. MATLAB uses the method based on the singular value decomposition, or SVD. The SVD algorithm is the most time consuming, but also the most reliable.
+
+The rank algorithm is:
+\begin{verbatim}
+s = svd(A);
+tol = max(size(A))*eps(max(s));
+r = sum(s > tol);
+\end{verbatim}
+
+See Also\cite{lapackuserguide1999}.
+
+
+
+
+
+\section{Null space of the matrix [TBD]}
+In linear algebra, the kernel or null space (also nullspace) of a matrix A is the set of all vectors $x$ for which $Ax = 0$. The kernel of a matrix with n columns is a linear subspace of n-dimensional Euclidean space. The dimension of the null space of A is called the nullity of A.
+
+
+\subsection{Computing of Null space}
+Numerical computation of null space
+
+Algorithms based on row or column reduction, that is, Gaussian elimination, presented in introductory linear algebra textbooks and in the preceding sections of this article are not suitable for a practical computation of the null space because of numerical accuracy problems in the presence of rounding errors. Namely, the computation may greatly amplify the rounding errors, which are inevitable in all but textbook examples on integers, and so give completely wrong results. For this reason, methods based on introductory linear algebra texts are generally not suitable for implementation in software; rather, one should consult contemporary numerical analysis sources for an algorithm like the one below, which does not amplify rounding errors unnecessarily.
+
+A state-of-the-art approach is based on singular value decomposition (SVD). This approach can be also easily programmed using standard libraries, such as LAPACK. SVD of matrix A computes unitary matrices U and V and a rectangular diagonal matrix S of the same size as A with nonnegative diagonal entries, such that
+
+$    \mathbf{U}\mathbf{S}\mathbf{V}^T = \mathbf{A}. $
+
+Denote the columns of V by (these are the rows of the transposed matrix $\mathbf{V}^T$ that appears in the decomposition)
+
+
+\paragraph{Null space in MATLAB}
+
+
+
+null
+
+
+Null space 
+Syntax 
+
+
+Z = null(A)
+Z = null(A,'r')
+
+Description
+
+
+Z = null(A) is an orthonormal basis for the null space of A obtained from the
+singular value decomposition. That is, A*Z has negligible elements, size(Z,2) is
+the nullity of A, and Z'*Z = I.
+
+Z = null(A,'r') is a "rational" basis for the null space obtained from the
+reduced row echelon form. A*Z is zero, size(Z,2) is an estimate for the nullity
+of A, and, if A is a small matrix with integer elements, the elements of the
+reduced row echelon form (as computed using rref) are ratios of small integers.
+
+The orthonormal basis is preferable numerically, while the rational basis may be
+preferable pedagogically.
+
+
+
+
+% 
+% 
+% \bibliographystyle{unsrt}  \bibliography{biblio/kmvlinalgdecompose}
+% \printindex
+% \end{document}