Introduction to matrix algebra | Free essay | Management essays

Introduction:

It is impossible to analyze statistics and peculiarly, econometrics without discoursing some most simple constructs of matrix algebra. Here the cardinal construct of matrix algebra is discussed for the better apprehension of the readers.

MATRIX ALZEBRA:

A matrix, allow it be denotes as ‘A ‘ is a rectangular array that consists of nm elements arranged in ‘n ‘ rows and ‘m ‘ columns. A typical component of A might be denoted by either Aij or aij, where I =1,2,3, … . , n and j = 1,2,3, … , m. The first inferior ever indicates the row, and the 2nd inferior indicates the column. It is sometimes necessary to demo the elements of a matrix explicitly, in which instance they are arrayed in rows and columns and surrounded by big baskets.

If a matrix has merely one column or merely one row, it is called as a ‘vector ‘ . There are two types of vectors, column vector and row vectors. Since column vectors are more common than row vectors, a vector that is non specified to be a row vector is usually treated as a column vector. If a column vector has ‘n ‘ elements, it is referred as an n-vector. Vectors like matrix are denoted with bold letters. On the other manus, if a matrix has the same figure of columns and rows, it is said as a ‘square vector ‘ . A square matrix A is a symmetric if Aij = Aji for all I and j. Symmetric matrices occur really often in practical applications of econometrics. A square matrix is said to be diagonal if Aij = 0 for all where the lone nonzero entries are those on what is called the chief diagonal. Sometimes a square matrix has all zero above or below the chief diagonal. Such a matrix is said to be triangular. If the nonzero elements are above the diagonal, it is said to be lower-triangular.

The transpose of a matrix is obtained by substituting its row and column inferiors. Thus the ijth component of becomes the jith component of its transpose, which is denoted as MT. It is seen that some research workers interpret the transpose as instead than MT. the transpose of a symmetric matrix is equal to the matrix itself. The transpose of a column vector is a row vector, and row vector will be the column vector.

A matrix is said to be symmetric merely if both the original matrix and its transpose are equal or on the other manus, matrix M is said to be symmetric if M =MT.

MATRIX MULTIPLICATION:

When two matrices are multiplies with each other, the ijth component of the new matrix is equal to the merchandise of the ith row of the first matrix with the jth column of the 2nd matrix. Here the basic consideration is that the first matrix ( allow M ) must hold m columns and that of the 2nd matrix ( allow P ) must hold ‘n ‘ rows. In general, if two matrices are to be comfy for generation, the first matrix must hold as many columns as the 2nd have equal sum of rows.

Again generation of two matrices is possible merely when they are of some common belongings. Give a matrix M of order m & A ; times ; Ns and another matrix P of order P & A ; times ; Qs, so the ordered brace ( M, P ) of the matrices M and P is said to be comfy if n = P, i.e. , the Numberss of columns of matrix M should be to the figure of rows of 2nd matrix P. Here the order of the matrix will be figure of elements of the first matrix ‘s row and 2nd matrix column.

Similarly, a matrix obtained by multiplying every component of the matrix ( P ) by R ( existent figure ) is called as the merchandise of P by R and is denoted by PR or RP.

Matrix generation is non commutative. Therefore, M & A ; times ; P? P & A ; times ; M where M and P are two matrices of same order.

An individuality matrix denoted as I is a diagonal matrix with every diagonal component equal to one ( 1 ) . This matrix is called as individuality matrix because when the matrix is pre-multiplied or station multiplied by any matrix, the original matrix remains unchanged. That is, for any matrix M, MI = IM = M.

MATRIX ADDITION:

There are different belongingss of matrix add-on. They are discussed as below.

  1. Associative– The associatory belongings of matrix add-on Tells that (Meter+Phosphorus) +K=Meter+ (Phosphorus+K) whereM = mij, P = Palestine Islamic JihadandK = kij
  2. Identity– Lashkar-e-taiba thatOmegabe am & A ; times ; Nsmatrix each of whose elements is of figure nothing, so 1.Meter+Omega=Omega+M = Z
  3. Inverse– when M be a given matrix ofm & A ; times ; Nsorder and P is another matrix with the samem & A ; times ; Nsorder and once more each component ofPhosphorusis the negative of the corresponding elements ofMeter, soM + P =P + M = Z.hereOmegais them x Norder zero matrix. The matrixPhosphorusis said to be negative ofMeterand is denoted by (-M) .
  4. Commutative– The matrix add-on is commutative in nature. In other wards, ifMeterandPhosphorusare two matrices ofm & A ; times ; Nsorder thanM+P=P+M.

MATRIX SUBTRACTION:

Let that Let thatMeterand andPhosphorusare two are twom & A ; times ; Nsmatrices, than matrix minus can be written as matrices, than matrix minus can be written as

M – P = M + ( -P ) .The difference between two matrices can be obtained by deducting each component of the first matrix to that of each corresponding elements of the 2nd matrix. Symbolically, sing two matrices M and P as derived above M-P.

Transpose OF THE PRODUCT OF TWO MATRICES:

The transpose of the merchandise of the two matrices is the merchandise of the transpose of the matrices with the order reversed. Let thatMeterandPhosphorusare two matrices of orderm & A ; times ; Ns.

Here the symbol ‘T ‘ or ‘ ” represent the same thing. The transpose of a matrix can be determined by substituting the rows into columns or columns into rows.

Adjoint of a Square Matrix:

The matrixMij, whereMijdenotes the co-factor of the original matrix i.e. , acquiring co-factor ofMijin the determiner of matrix, is called as the adjoint of matrixMeterand is denoted by the symboladj of Mso that the adjoint ofMeteris the response of the transpose of the matrix formed by the cofactors of matrixMeter.

There are two ways of ciphering the adjoint of a matrix.

  1. Get the transpose of the original matrix and than replace each component of the transpose matrix by ciphering the cofactors of the transpose matrix.
  2. Otherwise, one has to cipher the cofactors of each elements of a matrix and after acquiring the cofactors of each component, one has to acquire the transpose of the cofactor matrix.

After acquiring the value of the determiners, the several values are to be putted in the several topographic points. The new matrix will be the cofactor matrix. Adjoint can be determined by permuting the cofactor matrix.

INVERSE OF A SQUARE MATRIX:

MATRIX NOTATION IN CASE OF REGRESSION ANALYSIS:

As already discussed above, one of the greatest advantages of matrix algebra is that it is really easy to calculate. Hence each research worker wants to change over any mathematical equation into matrix signifier so that they can easy put to death the consequences. Such a procedure is the transmutation of arrested development theoretical accounts or equations into matrix notation. Following is the procedure of change overing a simple arrested development theoretical account into matrix signifier.

In a arrested development theoretical account, the value of X matrix is called as regressors or independent values and the column vector denoted by Y is the regressand or dependent variable. One greatest advantages of change overing the arrested development theoretical account into the matrix notation as derived above in an equation signifier is that here the Numberss of regressors are non restricted instead can be increased to any Numberss. Let see a instance there is holding thousand Numberss of regressors, one of which may or may non correspondent to a changeless and the others to a figure of explanatory variables.

The general rule of matrix generation is used while change overing the arrested development map into the equation signifier. These regulations make it possible to happen every bit convenient looks for other facets of arrested development theoretical accounts. The cardinal fact here lies is that every component of the merchandise of two matrices is a summing up. Thus it really convenient to utilize matrix algebra ( as discussed above in this chapter ) while covering with summing ups.

The other equation derived above is specific with three independent variables withithobservations, where as there is besides equation which is nil but the generalisation of‘m ‘independent variables. The independent variables are defines as,

X1i= theithobservations of the first independent variable

X2i= theithobservations of the 2nd independent variable and likewise

X3i= theithobservations of the 3rd independent variable.

The basic reading of the arrested development coefficient as derived in equation is that it shows the consequence of a one unit alteration inX1on independent variable Y, maintaining the other independent variables ( X2 and X3 ) as changeless. Like wise, in the equation measures the consequence of one unit alteration inX2on dependent variable Y, maintaining the other independent variables (X1 and X2) as changeless.

MATRIX NOTATION TO LEAST SQUARE:

One of the greatest advantages of the matrix attack is that it has the capableness to change over complicated theoretical jobs into simple one. In other wards, when a theoretical construct is converted into matrix notation, this helps the research workers to put to death the job faster and easier. Again, the least square estimation of arrested development analysis is holding its ain empirical utility in societal scientific discipline and direction surveies ( for item see chapter-14 of the book ) . But in instance of some multiple arrested development analysis, the construct becomes so complicated that it creates tonss of trouble to put to death. In similar state of affairss, the attacks need to be explained with the aid of algebra. The matrix algebra can be used to simplify the complicated theories by utilizing the method of least squares, the notations is more compact and provides theoretical insists every bit good as computational advantages. In least square appraisal, the parametric quantities are to be so chosen that it will minimise the mistake.

Decision:

Any complicated theoretical account, when converted into matrix notation can be solved easy. Matrix algebra as a tool of determination devising is holding monolithic usage in research. But the undertaking of change overing any theoretical account, peculiarly a complicated multiple arrested development theoretical account into a matrix notation is non so easy in world instead requires in-depth mathematical cognition.

Summary:

  1. A matrix is the rectangle array of rows and columns.
  2. If a matrix has merely one column or merely one row, it is called as a ‘vector ‘ .
  3. A square matrix A is a symmetric if Aij = Aji for all I and J.
  4. There are four of import belongingss of matrix add-on like Associative, Identity, Inverse and Commutative.
  5. The transpose of the merchandise of the two matrices is the merchandise of the transpose of the matrices with the order reversed.