Matlab matrix functions

Tn MATLAB, the function triu A) constructs an upper triangular matrix out of the matrix A, that is, it keeps the elements on the main diagonal and above that unchanged and replaces the elements located under the main diagonal with zero. 

In MATLAB, the function orth(A) gives the matrix Q described above. 

Note that in MATLAB, the functions exp(A), c. o.r(A), sin A), are element-by-element functions and do not obey the above definitions. The MATLAB functions expm A), expmJ A), expm2 A), and expmSiA) calculate exponential of the matrix A by different algorithms. The function expm2(A) calculates exponential of the matrix A as in Eq. (2.83). 

This tutorial looks at how MATLAB commands are used to convert transfer functions into state-space vector matrix representation, and back again. The discrete-time response of a multivariable system is undertaken. Also the controllability and observability of multivariable systems is considered, together with pole placement design techniques for both controllers and observers. The problems in Chapter 8 are used as design examples. 

We can check with MATLAB that the model matrix A has eigenvalues -0.29, -0.69, and -10.02. They are identical with the closed-loop poles. Given a block diagram, MATLAB can put the state space model together for us easily. To do that, we need to learn some closed-loop MATLAB functions, and we will defer this illustration to MATLAB Session 5. 

To compute the controllability matrix, we can use the MATLAB function ctrb () ... 

To evaluate the matrix polynomial in Eq. (9-23), we use the MATLAB function polyvalm () which applies the coefficients in p2 to the matrix A. 

Matlab is a matrix oriented language that is just about perfect for most data analysis tasks. Those readers who already know Matlab will agree with that statement. Those who have not used Matlab so far, will be amazed by the ease with which rather sophisticated programs can be developed. This strength of Matlab is a weak point in Excel. While Excel does include matrix operations, they are clumsy and probably for this reason, not well known and used. An additional shortcoming of Excel is the lack of functions for Factor Analysis or the Singular Value Decomposition. Nevertheless, Excel is very powerful and allows the analysis of fairly complex data. 

Matlab employs B( , 3) as the notation for the third column of B, b ,3. By using repmat (B ( , 3), 1,3) a matrix is created consisting of an l-by-3 (horizontal) tiling of copies of B( , 3). Naturally, this function can also be used to create a vertical tiling of copies of row vectors, e.g. if row vector b2, is to be added/subtracted to/from all rows of A. An appropriate function call would then be repmat (B (2, ), 2,1). We refer to the Matlab manuals for further details on this function. 

In contrast to Matlab, where the defaults are the matrix operators, in Excel the default is the element-wise operation. In fact, all basic operations (e-g- 0, 0> 0> 0> Q) and functions (e.g. EXP, LN, LOG) work element-wise in Excel. All... 

Within the Matlab s numerical precision X is singular, i.e. the two rows (and columns) are identical, and this represents the simplest form of linear dependence. In this context, it is convenient to introduce the rank of a matrix as the number of linearly independent rows (and columns). If the rank of a square matrix is less than its dimensions then the matrix is call rank-deficient and singular. In the latter example, rank(X)=l, and less than the dimensions of X. Thus, matrix inversion is impossible due to singularity, while, in the former example, matrix X must have had full rank. Matlab provides the function rank in order to test for the rank of a matrix. For more information on this topic see Chapter 2.2, Solving Systems of Linear Equations, the Matlab manuals or any textbook on linear algebra. 

Alternatively, Matlab s built-in function norm can be used to determine normalisation coefficients and perform the same task. An example for column-wise normalisation of a matrix X with orthogonal columns is given below. It is worthwhile to compare X with equation (2.15) the subspace command can be used to determine the angle between the vectors (in rad) and reconfirm orthogonality. ... 

Note that this kind of normalisation, via the norm function, can only be performed column- (or row-) wise via a loop as seen in the Matlab box above. Calling norm with one matrix argument determines a different kind of normalisation coefficients. We refer to the Matlab help and function references for more detail. 

Initially, we develop Matlab code and Excel spreadsheets for relatively simple systems that have explicit analytical solutions. The main thrust of this chapter is the development of a toolbox of methods for modelling equilibrium and kinetic systems of any complexity. The computations are all iterative processes where, starting from initial guesses, the algorithms converge toward the correct solutions. Computations of this nature are beyond the limits of straightforward Excel calculations. Matlab, on the other hand, is ideally suited for these tasks, as most of them can be formulated as matrix operations. Many readers will be surprised at the simplicity and compactness of well-written Matlab functions that resolve equilibrium systems of any complexity. 

This function is called numerous times from the Matlab ODE solver. In the example it is the ode45 which is the standard Runge-Kutta algorithm. ode45 requires as parameters the file name of the inner function, ode autocat. m, the vector of initial concentrations, cO, the rate constants, k, and the total amount of time for which the reaction should be modelled (20 time units in the example). The solver returns the vector t at which the concentrations were calculated and the concentrations themselves, the matrix C. Note that due to the adaptive step size control, the concentrations are computed at times t which are not predefined. 

Let us start with an example the Matlab function Data AB. m models the absorption spectra of a reacting solution as a function of time. They are stored as rows of the matrix Y. The reaction is a simple first order reaction A - B as introduced in Chapter 3.4.2, Rate Laws with Explicit Solutions. 

Matlab supplies the function Isqnonneg that performs a non-negative least-squares fit of the kind y=Ca+r, where y and a are column vectors. The function computes the best vector a with only positive entries. This equation corresponds to data acquired at only one wavelength. In our application, the columns of A have to be computed individually in a loop over all wavelengths, in each instance using the appropriate column of Y. C is the complete matrix of concentrations. It is, of course, the same for all wavelengths. 

In Constraint Positive Component Spectra (p.168), we introduced an improved, much faster matrix based function nonneg. m (provided by C. Andersson) that is more efficient than the Matlab function Isqnonneg. m. The result of implementing the function constra ints nonneg. m is identical but achieved much faster. 

Spectra taken during the Auger sputter profile were transformed into a data matrix and treated by the PHI-MATLAB , version 4.0, software. Each element was treated separately, that is the number of matrices was the same as the number of significant elements, and each matrix was completely independent of the others. The number of factors used for reconstruction was the one predicted by the minimum of the indicator function (IND). Unless specified, no mathematical treatment was performed on the data (differentiation, smoothing). 

Hint Linearize the equations and find the associated matrix eigenvalues by using the built-in MATLAB function eig.]... 

The function in Example 4.4 can be used to autoscale a data matrix. The function determines the size of the argument, its mean vector, and its standard deviation vector. On the last fine, a MATLAB programming trick is used to extend the mean vector and standard deviation vector into matrices having the same number of rows as the original argument prior to subtraction and division. The expression ones < r, i) creates an r x 1 column vector of ones. When used as an index in the statement mn (ones(r,1), ), it instructs MATLAB to replicate the mean vector r times to give a matrix having the dimensions r x c. 

C+ is the so-called pseudoinverse of C. It can be computed as C+ = (C C) 1 Cl. However, MATLAB provides a numerically superior method for the calculation of A by means of the back-slash operator ( ). Refer to the MATLAB manuals for details. The important point is that we are now in a position to write the residual matrix R, and thus ssq, as a function of the rate constants k only ... 

A significant advantage of Matlab is that there are several further veiy useful matrix operations. Most are in the form of functions the arguments are enclosed in brackets. Three that are important in chemometrics are as follows ... 

For a comprehensive list of facilities, see the manuals that come with Matlab, or the help hies however, a few that are useful to the reader of this book are as follows. The size function gives the dimensions of a matrix, so size (W) will return a 2 x 1 vector widi elements, in our example, of 2 and 3. It is possible to create a new vector, for example, s = size (W) in such a situation s (1) will equal 2, or the number of rows. The element W(s(l), s (2) ) represents the last element in die matrix W. In addition, it is possible to use die functions size(W,l) and size (W, 2) which provide die number of rows and columns directly. These functions are very useful when writing simple programs as discussed below. 

---