mf_matexpsym.hlp

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{smcl}
{* 28mar2005}{...}
{cmd:help mata matexpsym()}
{hline}
{* index exponentiation}{...}
{* index logarithms}{...}
{* index matexpsym()}{...}
{* index matlogsym()}{...}

{title:Title}

{p 4 4 2}
{bf:[M-5] matexpsym() -- Exponentiation and logarithms of symmetric matrices}


{title:Syntax}

{p 8 12 2}
{it:numeric matrix}
{cmd:matexpsym(}{it:numeric matrix A}{cmd:)}

{p 8 12 2}
{it:numeric matrix}
{cmd:matlogsym(}{it:numeric matrix A}{cmd:)}


{p 8 12 2}
{it:void}{bind:         }
{cmd:_matexpsym(}{it:numeric matrix A}{cmd:)}

{p 8 12 2}
{it:void}{bind:         }
{cmd:_matlogsym(}{it:numeric matrix A}{cmd:)}


{title:Description}

{p 4 4 2}
{cmd:matexpsym(}{it:A}{cmd:)} returns the matrix exponential of the
symmetric (Hermitian) matrix {it:A}.

{p 4 4 2}
{cmd:matlogsym(}{it:A}{cmd:)} returns the matrix natural logarithm of the
symmetric (Hermitian) matrix {it:A}.

{p 4 4 2}
{cmd:_matexpsym(}{it:A}{cmd:)} and {cmd:_matlogsym(}{it:A}{cmd:)} do the same
thing as {cmd:matexpsym()} and {cmd:matlogsym()}, but instead of returning the
result, they store the result in {it:A}.

{title:Remarks}

{p 4 4 2}
Do not confuse 
{cmd:matexpsym(}{it:A}{cmd:)}
with
{cmd:exp(}{it:A}{cmd:)}, 
nor 
{cmd:matlogsym(}{it:A}{cmd:)}
with
{cmd:log(}{it:A}{cmd:)}.  
{cmd:matexpsym(2*matlogsym(}{it:A}{cmd:))} produces the same result as 
{it:A}{cmd:*}{it:A}.
{cmd:exp()} and {cmd:log()} return elementwise results.

{p 4 4 2}
Exponentiated results and logarithms are obtained by extracting the eigenvalues
and eigenvectors of {it:A}, performing the operation on the eigenvalues, 
and then rebuilding the matrix.  That is, first {it:X} and {it:L} are found
such that

			{it:A}{it:X} = {it:X}*diag({it:L}){right:(1)   }

{p 4 4 2}
In the case of symmetric (Hermitian) matrix {it:A}, {it:X} is orthogonal,
meaning {it:X}'{it:X} = {it:X}{it:X}' = {it:I}.  Thus

			{it:A} = {it:X}*diag({it:L})*{it:X}'{right:(2)   }

{p 4 4 2}
{cmd:matexpsym(}{it:A}{cmd:)} is then defined

			{it:A} = {it:X}*diag(exp({it:L}))*{it:X}'{right:(3)   }

{p 4 4 2}
and {cmd:matlogsym(}{it:A}{cmd:)} is defined 

			{it:A} = {it:X}*diag(log({it:L}))*{it:X}'{right:(4)   }

{p 4 4 2}
(1) is obtained via {cmd:symeigensystem()}; see 
{bf:{help mf_eigensystem:[M-5] eigensystem()}}.



{title:Conformability}

    {cmd:matexpsym(}{it:A}{cmd:)}, {cmd:matlogsym(}{it:A}{cmd:)}:
		{it:A}:  {it:n x n}
	   {it:result}:  {it:n x n}

    {cmd:_matexpsym(}{it:A}{cmd:)}, {cmd:_matlogsym(}{it:A}{cmd:)}:
	{it:input:}
		{it:A}:  {it:n x n}
	{it:output:}
		{it:A}:  {it:n x n}

		
{title:Diagnostics}

{p 4 4 2}
{cmd:matexpsym(}{it:A}{cmd:)},
{cmd:matlogsym(}{it:A}{cmd:)},
{cmd:_matexpsym(}{it:A}{cmd:)}, 
and
{cmd:_matlogsym(}{it:A}{cmd:)}
return missing results if 
{it:A} contains missing values.

{p 4 4 2}
In addition:

{p 8 12 2}
1.
These functions do not check that {it:A} is symmetric or Hermitian. If {it:A}
is a real matrix, only the lower triangle, including the diagonal, is used.
If {it:A} is a complex matrix, only the lower triangle and the real parts of
the diagonal elements are used.

{p 8 12 2}
2.  
These functions return a matrix of the same storage type as {it:A}.  

{p 12 12 2}
In the case of {cmd:matlogsym(}{it:A}{cmd:)}, this means that if {it:A} is real
and the result cannot be expressed as a real, a matrix of missing values is
returned.  If you want the generalized solution, code
{cmd:matlogsym(C(}{it:A}{cmd:))}.  This is the same rule as with scalars:
{cmd:log(}-1{cmd:)} evaluates to missing, but {cmd:log(C(}-1{cmd:))} is
3.14159265i.

{p 8 12 2}
3.  
These functions are guaranteed to return a matrix that is numerically
symmetric, Hermitian, or symmetriconly if theory states that the matrix should
be symmetric, Hermitian, or symmetriconly.
See {bf:{help mf_matpowersym:[M-5] matpowersym()}} for a discussion of 
this issue.  

{p 12 12 2}
In the case of the functions here, real function exp({it:x}) is defined for
all real values of x (ignoring overflow), and thus the matrix returned by
{cmd:matexpsym()} will be symmetric (Hermitian).

{p 12 12 2}
The same is not true for {cmd:matlogsym()}.  log({it:x}{cmd:)} is not 
defined for {it:x}==0, so if any of the eigenvalues of {it:A} are 0 or 
very small, a matrix of missing values will result.  
In addition, 
log({it:x}) is complex for {it:x}<0, and thus if any of the eigenvalues are 
negative, the resulting matrix will 
be (1) missing if {it:A} is real stored as real, (2) symmetriconly if 
{it:A} contains reals stored as complex, and (3) general if {it:A} is complex.


{title:Source code}

{p 4 4 2}
{view matexpsym.mata, adopath asis:matexpsym.mata},
{view _matexpsym.mata, adopath asis:_matexpsym.mata},
{view matlogsym.mata, adopath asis:matlogsym.mata},
{view _matlogsym.mata, adopath asis:_matlogsym.mata},
{view _symmatfunc_work.mata, adopath asis:_symmatfunc_work.mata}

    
{title:Also see}

{p 4 13 2}
Manual:  {hi:[M-5] matexpsym()}

{p 4 13 2}
Online:  help for 
{bf:{help mf_matpowersym:[M-5] matpowersym()}},
{bf:{help mf_eigensystem:[M-5] eigensystem()}};
{bf:{help m4_matrix:[M-4] matrix}}
{p_end}