Manuals/calci/SYMMETRIC

MATRIX("SYMMETRIC",order)

• $order$ 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 $A^{T}=A$,where $A^{T}$ denotes the transpose.
• i.e., A square matrix which is equal to its transpose is called symmetric matrix.
• So $a_{ij}=a_{ji}$.
• This also implies $A^{-1}A^{T}=I$, 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

$A={\begin{pmatrix}43&-5&-93\\-5&-11&-75\\-93&-75&-7\\\end{pmatrix}}$

• 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

• PERSYMMETRIC
• PASCAL
• TRIANGULAR