Basic Operations of Matrix Algebra von Edu Pristine

Dozent des Vortrages Basic Operations of Matrix Algebra

Edu Pristine

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... from the corresponding element in the first. Resultant matrix obtained is of same order as that of matrix used for addition/subtraction If A = [a ij] and B = [b ij] are the two matrices of same order, say m*n, then the sum/difference of two matrices A and B is defined by matrix C = [c ij] , where c ij = a ij+ b ij. Example: Find A+B and A-B if ...

... Multiplication with a matrix. Matrices can also be multiplied together if the number of columns of the first matrix is equal to the number of rows of the second. For example, Multiplication would be defined on matrices A(mxn) and B(nxp) to get a resultant matrix C(mxp). If A = [a ij] be an m*n matrix and B = [b jk] are the two matrices of order n*p. Then, the product of matrices A and B is matrix C of order m*p. To get (i,k) element c ik of matrix C , we take i throw of A and k th column of B, multiply them element wise and take sum of all these product. Find AB if, multiplication of matrices is associative, that is A(BC) = (AB)C. Multiplication of matrices is not commutative, that is AB ...

... a matrix such that the product of the matrix and its inverse is an identity matrix. An identity matrix is one in which every entry is zero except for the main diagonal which contains ones. ...

... the portfolio, then Rp = wxR A+(1-w)xRB. Risk is measured by standard deviation, Risk (R) = S.D(Rp) = (Var(R p))1/2. For the two-asset portfolio A and B with weights w and (1-w) respectively ...

... Example: Consider a three asset portfolio with the following details: s 12 = 0.01234, s 22 = 0.02324, s 32 = 0.00946, s 12= 0.01435, s 23= 0.00654, s 13= 0.00750, w1 = 0.5, w2 = 0.4, w3 = 0.1. Calculate the volatility of the portfolio using matrix algebra = 0.0013911, S.D. = (0.0013911) = 0.1179 or 11.79% ...

... can also be expressed in the similar way as we expressed earlier. For quadratic forms matrix is 0.5 times of Hessian matrix (covered in Calculus). Involves quadratic terms such as x 2 , y 2 , z 2 , xy, yz, zx etc. Example: Variance involving three assets may involve terms such as w 1, w 2, w 12 , w 22 , w 32 ...

... Variance of quadratic forms: Sometimes quadratic forms are rewritten using the relationship between standard deviations (e.g., s 1 and s 2), the covariance (s 12), and the corresponding correlation coefficient (r 12). Relationship used is s 12= s 1s 2r 12 and can be generalised to matrix from using V = DCD. Putting the value of above matrix in 3 asset portfolio model we get ...