Numerical Methods in Computational Finance. Daniel J. Duffy

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target="_blank" rel="nofollow" href="#fb3_img_img_b97ef408-d624-52eb-a5e8-67153fc62e23.png" alt="StartLayout 1st Row 1 period Matrix addition left-parenthesis and subtraction right-parenthesis period 2nd Row 2 period Scalar multiplication period 3rd Row 3 period Matrix multiplication period 4th Row 4 period Matrix transpose period 5th Row 5 period Trace left-parenthesis s u m of diagonal elements of normal a matrix right-parenthesis period EndLayout"/>

       Matrix addition:

       Scalar multiplication:

       Matrix multiplication:

       Transpose of a matrix AA new matrix obtained by writing the rows of A as columns.Applicable to rectangular matrices:

      1 Matrix trace (the sum of the diagonal element of a square matrix):

      An example of matrix multiplication is:

Start 2 By 2 Matrix 1st Row 1st Column 1 2nd Column 3 2nd Row 1st Column 2 2nd Column negative 1 EndMatrix times Start 2 By 3 Matrix 1st Row 1st Column 17 2nd Column negative 6 3rd Column 14 2nd Row 1st Column negative 1 2nd Column 2 3rd Column negative 14 EndMatrix equals Start 2 By 3 Matrix 1st Row 1st Column 2 2nd Column 0 3rd Column negative 4 2nd Row 1st Column 5 2nd Column negative 2 3rd Column 6 EndMatrix period

z equals x plus italic i y comma z overbar equals x minus italic i y left-parenthesis italic complex conjugate of z right-parenthesis comma z comma z overbar element-of normal double struck upper C period

      In many cases we are concerned with square matrices with real values.

      5.6.1 Nilpotent and Related Matrices

      A nilpotent matrix A is a square matrix such that upper A Superscript p Baseline equals 0 for some positive integer p. It is said to be of index p if p the least positive integer for which upper A Superscript p Baseline equals 0. For example, the matrix:

upper A equals Start 2 By 2 Matrix 1st Row 1st Column 0 2nd Column 1 2nd Row 1st Column 0 2nd Column 0 EndMatrix

      is nilpotent with index 2. More generally, a triangular matrix of size n with zeros along the main diagonal is nilpotent with index less-than-or-equal-to n. For example, the follow matrix is nilpotent with index 3:

upper A equals Start 3 By 3 Matrix 1st Row 1st Column 1 2nd Column 5 3rd Column negative 2 2nd Row 1st Column 1 2nd Column 2 3rd Column negative 1 3rd Row 1st Column 3 2nd Column 6 3rd Column negative 3 EndMatrix period

      The determinant and trace of a nilpotent matrix are always zero. Thus, such matrices are not invertible. However, upper I minus upper N and upper I plus upper N are invertible where N is a nilpotent matrix:

      (5.16)StartLayout 1st Row left-parenthesis upper I minus upper N right-parenthesis Superscript negative 1 Baseline equals sigma-summation Underscript n equals 0 Overscript infinity Endscripts upper N Superscript n Baseline equals upper I plus upper N plus upper N squared plus ellipsis 2nd Row left-parenthesis upper I plus upper N right-parenthesis Superscript negative 1 Baseline equals sigma-summation Underscript n equals 0 Overscript infinity Endscripts left-parenthesis negative upper N right-parenthesis Superscript n Baseline equals left-parenthesis upper I minus left-parenthesis negative upper N right-parenthesis right-parenthesis Superscript negative 1 EndLayout

      where I is the identity matrix. Since there are only finitely many non-zero terms, we see that both sums converge.

      An idempotent matrix A is one for which upper A squared equals upper A. Examples are:

upper A equals Start 2 By 2 Matrix 1st Row 1st Column 1 2nd Column 0 2nd Row 1st Column 0 2nd Column 1 EndMatrix comma upper A equals Start 2 By 2 Matrix 1st Row 1st Column 3 2nd Column negative 6 2nd Row 1st Column 1 2nd Column negative 2 EndMatrix period

      Idempotent matrices arise in regression analysis and econometrics, for example in ordinary least squares problems, in particular when estimating sums of squared residuals.

upper A squared equals upper I

      An example is:

upper A equals Start 2 By 2 Matrix 1st Row 1st Column a 2nd Column b 2nd Row 1st Column c 2nd Column negative a EndMatrix provided a squared plus italic b c equals 1 period

      For example, the Pauli matrices are involutory:

StartLayout 1st Row 1st Column sigma 1 2nd Column equals 3rd Column sigma Subscript x 4th Column equals 5th Column Start 2 By 2 Matrix 1st Row 1st Column 0 2nd Column 1 2nd Row 1st Column 1 2nd Column 0 EndMatrix 2nd Row 1st Column sigma 2 2nd Column equals 3rd Column sigma Subscript y 4th Column equals 5th Column Start 2 By 2 Matrix 1st Row 1st Column 0 2nd Column negative i 2nd Row 1st Column i 2nd Column 0 EndMatrix left-parenthesis i equals StartRoot negative 1 EndRoot right-parenthesis 3rd Row 1st Column sigma 3 2nd Column equals 3rd Column sigma Subscript z 4th Column equals 5th Column Start 2 By 2 Matrix 1st Row 1st Column 1 2nd Column 0 2nd Row 1st Column 0 2nd Column negative 1 EndMatrix period EndLayout

      5.6.2 Normal Matrices

      We introduce the important class of normal matrices by introducing some prerequisite notation. The transpose of a real m times n matrix is an n times m matrix formed by exchanging the rows and columns of A:

StartLayout 1st Row upper A equals left-parenthesis a Subscript italic i j Baseline right-parenthesis comma 1 less-than-or-equal-to i less-than-or-equal-to m comma 1 less-than-or-equal-to j less-than-or-equal-to m 2nd Row upper A Superscript down-tack Baseline equals left-parenthesis a Subscript italic j i Baseline right-parenthesis comma 1 less-than-or-equal-to j less-than-or-equal-to n comma 1 less-than-or-equal-to i less-than-or-equal-to m left-parenthesis 
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