Recursive Estimation and Time-Series Analysis An Introduction
This book has grown out of a set of lecture notes prepared originally for a NATO Summer School on "The Theory and Practice of Systems ModelLing and Identification" held between the 17th and 28th July, 1972 at the Ecole Nationale Superieure de L'Aeronautiq
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A.ll MatrixAlgebra 1. Matri ces
columns;
A matrix is defined as a rectangular array of elements arranged in rows and in this book it is denoted by a capital letter, e.g. au a 12
a 1n
a 21 a22
a2n
A=
Often A is alternatively denoted elements aij , i = 1, 2, ... , m; in m rows and n columns, then it The following should be (i)
by [a ij ] to indicate that it is characterized by j = 1, 2, ... , n. If it has m.n elements arranged is said to be of order m by n, usually written m x n. noted in relation to matrices:
a nuZZ matrix has all of its elements set to zero, i.e. aij = 0 for all i, j;
a symmetric matrix is a square matrix in which aij = aji ; i.e. it is symmetric about the diagonal elements; (iii) the trace of a square n x n matrix, denoted by Tr., is the sum of its diagonal elements i.e. Tr.A = all + a 22 + .... + ann
(ii)
(iv)
a diagonaZ matrix is a square matrix with all its elements except those on the diagonaZ set to zero i.e.
A=
o
246
(v)
an n x n diagonal matrix with elements set to unity is denoted by In and termed the identity (or unit) matrix of order n, e.g. for a 3 x 3 identity matri x
sometimes the subscript is omitted if the order is obvious. (vi) an idempotent matrix is a square matrix such that A2
=
AA
=
A
i.e. it remains unchanged when multiplied by itself. 2. Vectors A matrix of order m x I contains a single column of m elements and is termed a coZumn vector (or sometimes just a vector); in this book, it is denoted by a lower case letter with an underscore i.e. for a vector b
b =
. 3. Matrix Addition (or Subtraction) If two matrices A and B are of the same order then we define A + B to be a new matrix C where cij = aij + bij In other words, the addition of the matrices is accomplished by adding corresponding elements. A - B is defined in an analogous manner. 4. Matrix or Vector Transpose The transpose of a matrix A is obtained from A by interchanging the rows and columns; in this book, it is denoted by a superscript capital T; e.g. for A defined in 1., above, au a2I amI aI2 a22
am2
247
The transpose of a column vector ~, denoted by ~T is termed a row vector, e.g. for bin 2., above, Note that (i) in the case of a symmetric matrix AT (ii) [ATl T = A ( iii) [A+B 1T = AT + BT
=
A
5. Matrix Multiplication
If A is of order m x nand B is of order n x p then the product AB is defined to be a matrix of order m x p whose (ij)th element cij is given by n
cij
= k~l aik bkj
i.e. the (ij)th element is obtained by, in turn, multiplying the elements of the ith row of the matrix A by the jth column of the matrix B and summing over all terms (:. the number of elements (n) in each row of A must be equal to the number of elements in each column of B). Note that, in general, the commutative law of multiplication which applies for scalars does not apply for matrices i.e. AB 1. BA so that pre-multiplication of B by A does not, in general, yield the same as postmUltiplication of B by A. However, pre-multiplying or post-multiplying by the identity matrix leaves the matrix unchanged i.
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