Manuals/calci/SYMMETRIC

MATRIX("SYMMETRIC",order)


  • is the size of the Symmetric matrix.

Description

  • This function returns the symmetric matrix of order 3.
  • A symmetric matrix is a square matrix that satisfies  ,where   denotes the transpose.
  • i.e., A square matrix which is equal to its transpose is called symmetric matrix.
  • So  .
  • This also implies  , where I is the identity matrix.
  • Because equal matrices have equal dimensions, only square matrices can be symmetric.
  • An example for the symmetric matrix is

 

  • The properties of symmetric matrices are:
  • 1.Every square diagonal matrix is symmetric, since all off-diagonal entries are zero.
  • 2.Similarly, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative.
  • 3.Hermitian matrices are a useful generalization of symmetric matrices for complex matrices.
  • In Calci, MATRIX("symmetric") gives the symmetric matrix with the integer numbers.
  • The other way to give the syntax is MATRIX("symmetric:integer).
  • The syntax is to get the positive numbers symmetric matrix is MATRIX("symmetric:positive integer").
  • To get a negative numbers symmetric matrix is MATRIX("symmetric:negative integer").
  • Also to get the symmetric matrix with the elements 0 and 1(boolean numbers) users give syntax as MATRIX("symmetric:boolean").
  • So using Calci users can get a different types of symmetric matrices.

Examples

  • 1.MATRIX("symmetric")
-10 88 92
88 14 -21
92 -21 -29
  • 2.MATRIX("symmetric:boolean",4)
1 0 1 1
0 0 1 0
1 1 0 1
1 0 1 1
  • 3.MATRIX("symmetric:integer",5)
-76 -15 7 -100 -28
-15 -32 -98 -100 -87
7 -98 47 52 -72
-100 -100 52 -63 8
-28 -87 -72 8 76

See Also

References