mf_eigensystem.hlp

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{smcl}
{* 25mar2005}{...}
{cmd:help mata eigensystem()}
{hline}
{* index eigenvectors}{...}
{* index eigenvalues}{...}
{* index characteristic roots}{...}
{* index latent roots}{...}
{* index proper values}{...}
{* index right eigenvectors}{...}
{* index left eigenvectors}{...}
{* index eigensystem()}{...}
{* index eigenvalues()}{...}
{* index symeigensystem()}{...}
{* index symeigenvalues()}{...}
{* index lefteigensystem()}{...}
{* index _eigensystem()}{...}
{* index _eigenvalues()}{...}
{* index _symeigensystem()}{...}
{* index _symeigenvalues()}{...}
{* index _lefteigensystem()}{...}
{* index _eigen_la()}{...}
{* index _symeigen_la()}{...}
{* index LAPACK}{...}

{title:Title}

{p 4 4 2}
{bf:[M-5] eigensystem() -- Eigenvalues and eigenvectors}


{title:Syntax}

{p 8 40 2}
{it:void}{bind:                  }
{cmd:eigensystem(}{it:A}{cmd:,} 
{it:X}{cmd:,} 
{it:L} 
[{cmd:,} {it:rcond} 
[{cmd:,} {it:nobalance}]]{cmd:)}

{p 8 40 2}
{it:void}{bind:              }
{cmd:lefteigensystem(}{it:A}{cmd:,} {it:X}{cmd:,} 
{it:L}
[{cmd:,} {it:rcond}
[{cmd:,} {it:nobalance}]]{cmd:)}

{p 8 40 2}
{it:complex rowvector}{bind:     }
{cmd:eigenvalues(}{it:A}{bind:      }
[{cmd:,} {it:rcond}
[{cmd:,} {it:nobalance}]]{cmd:)}


{p 8 40 2}
{it:void}{bind:               }
{cmd:symeigensystem(}{it:A}{cmd:,} {it:X}{cmd:,} {it:L}{cmd:)}

{p 8 40 2}
{it:real rowvector}{bind:     }
{cmd:symeigenvalues(}{it:A}{cmd:)}


{p 8 40 2}
{it:void}{bind:                 }
{cmd:_eigensystem(}{it:A}{cmd:,} 
{it:X}{cmd:,} 
{it:L} 
[{cmd:,} {it:rcond} 
[{cmd:,} {it:nobalance}]]{cmd:)}

{p 8 40 2}
{it:void}{bind:             }
{cmd:_lefteigensystem(}{it:A}{cmd:,} {it:X}{cmd:,} 
{it:L}
[{cmd:,} {it:rcond}
[{cmd:,} {it:nobalance}]]{cmd:)}

{p 8 40 2}
{it:complex rowvector}{bind:    }
{cmd:_eigenvalues(}{it:A}{bind:      }
[{cmd:,} {it:rcond} 
[{cmd:,}
{it:nobalance}]]{cmd:)}

{p 8 40 2}
{it:void}{bind:              }
{cmd:_symeigensystem(}{it:A}{cmd:,} {it:X}{cmd:,} {it:L}{cmd:)}

{p 8 40 2}
{it:real rowvector}{bind:    }
{cmd:_symeigenvalues(}{it:A}{cmd:)}



{p 4 4 2}
where inputs are 

			{it:A}:  {it:numeric matrix}
		    {it:rcond}:  {it:real scalar} (whether {it:rcond} desired)
		{it:nobalance}:  {it:real scalar} (whether to suppress balancing)

{p 4 4 2}
and outputs are

			{it:X}:  {it:numeric matrix} of eigenvectors 
			{it:L}:  {it:numeric vector} of eigenvalues
		    {it:rcond}:  {it:real vector} of reciprocal condition numbers

{p 4 4 2}
The columns of {it:X} will contain the eigenvectors except when using 
{cmd:_lefteigensystem()}, in which case the rows of {it:X} contain the 
eigenvectors.


{p 4 4 2}
The following routines are used in implementing the above routines:

{p 8 30 2}
{it:real scalar}
{cmd:_eigen_la(}{it:real scalar todo}{cmd:,}
{it:numeric matrix A}{cmd:,}
{it:X}{cmd:,}
{it:L}{cmd:,}
{it:real scalar rcond}{cmd:,}
{it:real scalar nobalance}{cmd:)}

{p 8 33 2}
{it:real scalar}
{cmd:_symeigen_la(}{it:real scalar todo}{cmd:,}
{it:numeric matrix A}{cmd:,}
{it:X}{cmd:,}
{it:L}{cmd:)}

		
{title:Description}

{p 4 4 2}
These routines calculate eigenvalues and eigenvectors of square matrix {it:A}.

{p 4 4 2}
{cmd:eigensystem(}{it:A}{cmd:,} {it:X}{cmd:,} {it:L}{cmd:,}
{it:rcond}{cmd:,} {it:nobalance}{cmd:)} 
calculates eigenvalues and eigenvectors of a general, real or complex,
square matrix
{it:A}.  Eigenvectors are returned in {it:X} and eigenvalues in {it:L}.
The remaining arguments are optional:

{p 8 12 2}
    1.  If {it:rcond} is not specified, then reciprocal condition numbers are
        not returned in {it:rcond}.

{p 12 12 2}
        If {it:rcond} is specified and contains a value other than 0 or
        missing -- {it:rcond}=1 is suggested -- in {it:rcond} will be placed a
        vector of the reciprocals of the condition numbers for the
        eigenvalues.  Each element of the new {it:rcond} measures the accuracy
        to which the corresponding eigenvalue has been calculated; large
        numbers (numbers close to 1) are better and small numbers (numbers
        close to 0) indicate inaccuracy; 
        see {it:Eigenvalue condition} below.

{p 8 12 2} 
    2.  If {it:nobalance} is not specified, balancing is
        performed in order to obtain more accurate results.

{p 12 12 2}
        If {it:nobalance} is specified and is not zero nor missing, 
        balancing is not used.  Results are calculated more quickly, but
        perhaps a little less accurately; see {it:Balancing} below.

{p 4 4 2}
{cmd:lefteigensystem(}{it:A}{cmd:,} {it:X}{cmd:,} {it:L}{cmd:,}
{it:rcond}{cmd:,} {it:nobalance}{cmd:)} mirrors 
{cmd:eigensystem()}, the difference being that {cmd:lefteigensystem()}
solves for left eigenvectors solving 
{it:X}{it:A}=diag({it:L})*{it:X}
instead of right eigenvectors solving
{it:A}{it:X}={it:X}*diag({it:L}).

{p 4 4 2}
{cmd:eigenvalues(}{it:A}{cmd:,} {it:rcond}{cmd:,} {it:nobalance}{cmd:)}
returns the eigenvalues of square matrix {it:A}; the eigenvectors are not 
calculated.  Arguments {it:rcond} and {it:nobalance} are optional.

{p 4 4 2}
{cmd:symeigensystem(}{it:A}{cmd:,} {it:X}{cmd:,} {it:L}{cmd:)}
and 
{cmd:symeigenvalues(}{it:A}{cmd:)} mirror {cmd:eigensystem()} and
{cmd:eigenvalues()}, the difference being that {it:A} is assumed to be
symmetric (Hermitian).  The eigenvalues returned are real.  (Arguments
{it:rcond} and {it:nobalance} are not allowed; {it:rcond}, because 
symmetric matrices are inherently well conditioned; {it:nobalance}, 
because it is unnecessary.)

{p 4 4 2}
The underscore routines mirror the routines of the same name without 
underscores, the difference being that {it:A} is damaged during the 
calculation and so the underscore routines use less memory.

{p 4 4 2}
{cmd:_eigen_la()} and {cmd:_symeigen_la()} are the interfaces into the
{bf:{help m1_lapack:[M-1] LAPACK}}
routines used to implement the above functions.  Their direct use is not 
recommended.


{title:Remarks}

{p 4 4 2}
Remarks are presented under the headings

	{bf:Eigenvalues and eigenvectors}
	{bf:Left eigenvectors}
	{bf:Symmetric eigensystems}
	{bf:Normalization and order}
	{bf:Eigenvalue condition}
	{bf:Balancing}
	{bf:eigensystem() and eigenvalues()}
	{bf:lefteigensystem()}
	{bf:symeigensystem() and symeigenvalues()}


{title:Eigenvalues and eigenvectors}

{p 4 4 2}
A scalar {it:l} (usually denoted by {it:lambda}) is said to be an
eigenvalue of square matrix {it:A}: {it:n x n} if there is a nonzero column
vector {it:x}: {it:n x} 1 (called the eigenvector) such that

			{it:A}{it:x} = {it:l}{it:x}{right:(1)   }

{p 4 4 2}
(1) can also be written 

			({it:A} - {it:l}{it:I}){it:x} = 0

{p 4 4 2}
where {it:I} is the {it:n x n} identity matrix.
A nontrivial solution to this system of n linear homogeneous equations 
exists if and only if 

			det({it:A} - {it:l}{it:I}) = 0{right:(2)   }

{p 4 4 2}
This {it:n}th degree polynomial in {it:l} is called the characteristic 
polynomial or characteristic equation of {it:A}, and the eigenvalues {it:l}
are its roots, also known as the characteristic roots.

{p 4 4 2}
There are, in fact, {it:n} solutions ({it:l_i}, {it:x_i}) that satisfy (1) --
although some can be repeated -- and we can compactly write the full set of
solutions as

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

{p 4 4 2}
where 

			{it:X} = ({it:x}_1, {it:x}_2, ...){...}
{col 60}({it:X}: {it:n x n})

			{it:L} = ({it:l}_1, {it:l}_2, ...){...}
{col 60}({it:L}: 1 {it:x n})

{p 4 4 2}
For instance, 


	: {cmd:A = (1, 2 \ 9, 4)}
	: {cmd:X = .}
	: {cmd:L = .}
	: {cmd:eigensystem(A, X, L)}

	: {cmd:X}
	{res}       {txt}          1             2
	    {c TLC}{hline 29}{c TRC}
	  1 {c |}  {res}-.316227766   -.554700196{txt}  {c |}
	  2 {c |}  {res}-.948683298    .832050294{txt}  {c |}
	    {c BLC}{hline 29}{c BRC}

	: {cmd:L}
	{res}       {txt} 1    2
	    {c TLC}{hline 11}{c TRC}
	  1 {c |}  {res} 7   -2{txt}  {c |}
	    {c BLC}{hline 11}{c BRC}

{p 4 4 2}
The first eigenvalue is 7, and the corresponding eigenvector is 
(-.316 \ -.949).
The second eigenvalue is -2, and the corresponding eigenvector is 
(-.555 \ .832).

{p 4 4 2}
In general, eigenvalues and vectors can be complex even if {it:A} is real.


{title:Left eigenvectors}

{p 4 4 2}
What we have defined above is properly known as the right-eigensystem 
problem:

			{it:A}{it:x} = {it:l}{it:x}{right:(1)   }

{p 4 4 2}
In the above, {it:x} is a column vector.  The left-eigensystem problem 
is to find the row vector {it:x} satisfying

			{it:x}{it:A} = {it:l}{it:x}{right:(1')  }

{p 4 4 2}
The eigenvalue {it:l} is the same in (1) and (1'), but {it:x} can differ.

{p 4 4 2}
The {it:n} solutions ({it:l_i}, {it:x_i}) that satisfy (1') 
can be compactly written,

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

{p 4 4 2}
where 

                     {c TLC}     {c TRC}                 {c TLC}     {c TRC}
		     {c |} {it:x}_1 {c |}                 {c |} {it:l}_1 {c |}
		     {c |} {it:x}_2 {c |}                 {c |} {it:l}_2 {c |}
	       X  =  {c |}  .  {c |}           L  =  {c |}  .  {c |}
		     {c |}  .  {c |}                 {c |}  .  {c |}
		     {c |} {it:x}_{it:n} {c |}                 {c |} {it:l}_{it:n} {c |}
                     {c BLC}     {c BRC}                 {c BLC}     {c BRC}
                     ({it:n x n})                 ({it:n x} 1)

{p 4 4 2}
For instance, 

	: {cmd:A = (1, 2 \ 9, 4)}
	: {cmd:X = .}
	: {cmd:L = .}
	: {cmd:lefteigensystem(A, X, L)}

	: {cmd:X}
	{res}       {txt}          1             2
	    {c TLC}{hline 29}{c TRC}
	  1 {c |}  {res}-.832050294   -.554700196{txt}  {c |}
	  2 {c |}  {res}-.948683298    .316227766{txt}  {c |}
	    {c BLC}{hline 29}{c BRC}

	: {cmd:L}
	{res}       {txt} 1
	    {c TLC}{hline 6}{c TRC}
	  1 {c |}  {res} 7{txt}  {c |}
	  2 {c |}  {res}-2{txt}  {c |}
	    {c BLC}{hline 6}{c BRC}

{p 4 4 2}
The first eigenvalue is 7, and the corresponding eigenvector is 
(-.832, -.555).
The second eigenvalue is -2, and the corresponding eigenvector is 
(-.949, .316).

{p 4 4 2}
The eigenvalues are the same as in the previous example; the eigenvectors 
are different.


{title:Symmetric eigensystems}

{p 4 4 2}
Below we use the term symmetric to encompass Hermitian matrices, even 
when we do not emphasize the fact.

{p 4 4 2}
Eigensystems of symmetric matrices are conceptually no different from general
eigensystems, but symmetry introduces certain simplifications:

{p 8 12 2}
    1.  The eigenvalues associated with symmetric matrices are real,
        whereas those associated with general matrices may be real
        or complex.

{p 8 12 2}
    2.  The eigenvectors associated with symmetric matrices -- which 
        may be real or complex -- are orthogonal.

{p 8 12 2}
    3.  The left and right eigenvectors of symmetric matrices 
        are transposes of each other.  

{p 8 12 2}
    4.  The eigenvalues and eigenvectors of symmetric matrices are 
        more easily, and more accurately, computed.

{p 4 4 2}