+
    i                     v    R t ^ RIHt ^ RIHt ^ RIHt R tR t	RR lt
]! R4      t]! RRR	R
7      R 4       tR# )z!Known matrices related to physics)I)MutableDenseMatrix)
deprecatedc                    V ^8X  d   RpM.V ^8X  d   ^ \         ) 3\         ^ 33pMV ^8X  d   RpM\        R4      h\        V4      # )zReturns a Pauli matrix `\sigma_i` with `i=1,2,3`.

References
==========

.. [1] https://en.wikipedia.org/wiki/Pauli_matrices

Examples
========

>>> from sympy.physics.matrices import msigma
>>> msigma(1)
Matrix([
[0, 1],
[1, 0]])
zInvalid Pauli index))       r   r   )r   )r   )r   
IndexErrorMatrix)imats   & v/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/physics/matrices.pymsigmar      sZ    " 	Av
 
aGF
 
a

 .//#;    c                    V) V,          pV) V,          pV) V,          pV^,          pV^,          pV^,          p	W,           WF3WGV	,           V3WeW,           33p
V \        V
4      ,          # )ag  Returns the Parallel Axis Theorem matrix to translate the inertia
matrix a distance of `(dx, dy, dz)` for a body of mass m.

Examples
========

To translate a body having a mass of 2 units a distance of 1 unit along
the `x`-axis we get:

>>> from sympy.physics.matrices import pat_matrix
>>> pat_matrix(2, 1, 0, 0)
Matrix([
[0, 0, 0],
[0, 2, 0],
[0, 0, 2]])

)r   )mdxdydzdxdydydzdzdxdxdxdydydzdzr   s   &&&&       r   
pat_matrixr   -   sw    $ 3r6D3r6D3r6Dq5Dq5Dq5DK$+t$$&C VC[=r   c                   V R9  d   \        R4      hV ^ 8X  d   RpMMV ^8X  d   RpMCV ^8X  d*   ^ ^ ^ \        ) 3^ ^ \        ^ 3^ \        ^ ^ 3\        ) ^ ^ ^ 33pMV ^8X  d   RpM	V ^8X  d   Rp\        X4      pV'       d   V R9   d   V) pV# )a@  Returns a Dirac gamma matrix `\gamma^\mu` in the standard
(Dirac) representation.

Explanation
===========

If you want `\gamma_\mu`, use ``gamma(mu, True)``.

We use a convention:

`\gamma^5 = i \cdot \gamma^0 \cdot \gamma^1 \cdot \gamma^2 \cdot \gamma^3`

`\gamma_5 = i \cdot \gamma_0 \cdot \gamma_1 \cdot \gamma_2 \cdot \gamma_3 = - \gamma^5`

References
==========

.. [1] https://en.wikipedia.org/wiki/Gamma_matrices

Examples
========

>>> from sympy.physics.matrices import mgamma
>>> mgamma(1)
Matrix([
[ 0,  0, 0, 1],
[ 0,  0, 1, 0],
[ 0, -1, 0, 0],
[-1,  0, 0, 0]])
zInvalid Dirac index)r   r            )r   r   r   r   r   r   r   r   r   r   r	   r   r   r   r   r	   )r   r   r   r   r   r   r   r   r   r	   r   r   r	   r   r   r   )r&   r$   r(   r"   )r&   r%   r!   r"   )r   r   r   r    )r
   r   r   )mulowerr   r   s   &&  r   mgammar+   K   s    > 
 .//	Qw
 
q
 
q1qbM1aL1aLRAqM	
 
q
 
q
 	sAAHr   zk
    The sympy.physics.matrices.mdft method is deprecated. Use
    sympy.DFT(n).as_explicit() instead.
    z1.9zdeprecated-physics-mdft)deprecated_since_versionactive_deprecations_targetc                :    ^ RI Hp V! V 4      P                  4       # )z
.. deprecated:: 1.9

   Use DFT from sympy.matrices.expressions.fourier instead.

   To get identical behavior to ``mdft(n)``, use ``DFT(n).as_explicit()``.
)DFT)"sympy.matrices.expressions.fourierr/   
as_mutable)nr/   s   & r   mdftr3      s      7q6r   N)F)r!   r'   r#   r$   )__doc__sympy.core.numbersr   sympy.matrices.denser   r   sympy.utilities.decoratorr   r   r   r+   minkowski_tensorr3    r   r   <module>r:      s[    '   = 0"J<HX      #8		r   