Manuals/calci/HERMITIAN

MATRIX("HERMITIAN",order)

• $order$ is the order of the Hermitian matrix.

Description

• This function gives the Hermitian matrix of order 3.
• A Hermitian matrix is defined as the square matrix with complex entries which is equal to its own conjugate transpose.
• i.e., the matrix A is Hermitian if and only if $A=A^{T}$, where $A^{T}$ denotes the conjugate transpose, which is equivalent to the condition Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{ij}=a^_{ji}} .
• A hermetian matrix is also called as self-adjoint matrix.
• The following matrix is the example of 3x3 Hermitian matrix:

${\begin{bmatrix}2&2+i&4\\2-i&3&i\\4&-i&1\\\end{bmatrix}}$.

• The diagonal elements must be real, as they must be their own complex conjugate.
• An integer or real matrix is Hermitian iff it is symmetric.
• In calci, users can change the order and number of the Hermitian matrices.

Examples

• 1.MATRIX("hermitian")

-62	-48 + 4i	49 + -40i
-48 + -4i	-54	0 + 34i
49 + 40i	0 + -34i	-33

• 2.MATRIX("hermitian",5)

-90	-75 + 79i	56 + -17i	92 + -51i	-13 + -21i
-75 + -79i	-19	-77 + -19i	42 + 47i	83 + -95i
56 + 17i	-77 + 19i	-60	-25 + -26i	88 + -81i
92 + 51i	42 + -47i	-25 + 26i	-89	-70 + -92i
-13 + 21i	83 + 95i	88 + 81i	-70 + 92i	-7

See Also

• ANTIDIAGONAL
• CONFERENCE
• CIRCULANT
• HANKEL