spss sYntax to get eigenvalues

[asghar]

hello to all

I want to estimate eigenvalues of non-symmetric matrices using SPSS syntax. below matrix is an example. can anyone tell me how we can get eigenvalues of a non-symmetric matrix in SPSS syntax.

MATRIX.
COMP S={0.2273, -0.0852;
             -0.1930,0.1794}.
CALL EIGEN(S,VECTOR,VALUE).
PRINT VECTOR.
PRINT VALUE.
END MATRIX.

with best regards,
Asghar Minaei

[David Marso]

MATRIX.
COMPUTE S={0.2273, -0.0852;
             -0.1930,0.1794}.
CALL SVD (S,U,Q,V).
PRINT {U,Q,V}.
PRINT  (S-(U * Q * T(V))) /FORMAT "F20.18".
END MATRIX.

From the FM:
"SVD(M,var1,var2,var3). Singular value decomposition of a matrix. Takes four arguments: a matrix and three valid variable names to which the results are assigned. If M is a matrix, the statement CALL SVD(M,U,Q,V) will assign to Q a diagonal matrix of the same dimensions as M, and to U and V unitary matrices (matrices whose inverses equal their transposes) of appropriate dimensions, such that M=U*Q* T(V), where T is the transpose function defined above. The singular values of M are in the main diagonal of Q."

asghar wrote

hello to all

I want to estimate eigenvalues of non-symmetric matrices using SPSS syntax. below matrix is an example. can anyone tell me how we can get eigenvalues of a non-symmetric matrix in SPSS syntax.

MATRIX.
COMP S={0.2273, -0.0852;
             -0.1930,0.1794}.
CALL EIGEN(S,VECTOR,VALUE).
PRINT VECTOR.
PRINT VALUE.
END MATRIX.

with best regards,
Asghar Minaei

[Kirill Orlov]

David, nope, the OP wants eigendecomposition of the asymmetric matrix which isn't the same as SVD.
SPSS has no such a function.

21.11.2013 0:18, David Marso пишет:

MATRIX.
COMPUTE S={0.2273, -0.0852;
             -0.1930,0.1794}.
CALL SVD (S,U,Q,V).
PRINT {U,Q,V}.
PRINT  (S-(U * Q * T(V))) /FORMAT "F20.18".
END MATRIX.

>From the FM:
"SVD(M,var1,var2,var3). Singular value decomposition of a matrix. Takes four
arguments: a matrix and three valid variable names to which the results are
assigned. If M is a matrix, the statement CALL SVD(M,U,Q,V) will assign to Q
a diagonal matrix of the same dimensions as M, and to U and V unitary
matrices (matrices whose inverses equal their transposes) of appropriate
dimensions, such that M=U*Q* T(V), where T is the transpose function defined
above. The singular values of M are in the main diagonal of Q."


asghar wrote

hello to all

I want to estimate eigenvalues of non-symmetric matrices using SPSS
syntax. below matrix is an example. can anyone tell me how we can get
eigenvalues of a non-symmetric matrix in SPSS syntax.

MATRIX.
COMP S={0.2273, -0.0852;
             -0.1930,0.1794}.
CALL EIGEN(S,VECTOR,VALUE).
PRINT VECTOR.
PRINT VALUE.
END MATRIX.

with best regards,
Asghar Minaei

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[David Marso]

Kirill,
Please enlighten the class as to how an SVD is NOT the same as an eigendecomposition.
In my example Q are eigenvalues, U and V are orthonormal .
If one does an SVD of a symmetrical matrix the results are identical to EIGEN.

Kirill Orlov wrote

David, nope, the OP wants eigendecomposition of the asymmetric matrix
which isn't the same as SVD.
SPSS has no such a function.

21.11.2013 0:18, David Marso ?????:
> MATRIX.
> COMPUTE S={0.2273, -0.0852;
>               -0.1930,0.1794}.
> CALL SVD (S,U,Q,V).
> PRINT {U,Q,V}.
> PRINT  (S-(U * Q * T(V))) /FORMAT "F20.18".
> END MATRIX.
>
> >From the FM:
> "SVD(M,var1,var2,var3). Singular value decomposition of a matrix. Takes four
> arguments: a matrix and three valid variable names to which the results are
> assigned. If M is a matrix, the statement CALL SVD(M,U,Q,V) will assign to Q
> a diagonal matrix of the same dimensions as M, and to U and V unitary
> matrices (matrices whose inverses equal their transposes) of appropriate
> dimensions, such that M=U*Q* T(V), where T is the transpose function defined
> above. The singular values of M are in the main diagonal of Q."
>
>
> asghar wrote
>> hello to all
>>
>> I want to estimate eigenvalues of non-symmetric matrices using SPSS
>> syntax. below matrix is an example. can anyone tell me how we can get
>> eigenvalues of a non-symmetric matrix in SPSS syntax.
>>
>> MATRIX.
>> COMP S={0.2273, -0.0852;
>>               -0.1930,0.1794}.
>> CALL EIGEN(S,VECTOR,VALUE).
>> PRINT VECTOR.
>> PRINT VALUE.
>> END MATRIX.
>>
>> with best regards,
>> Asghar Minaei
>
>
>
> -----
> Please reply to the list and not to my personal email.
> Those desiring my consulting or training services please feel free to email me.
> ---
> "Nolite dare sanctum canibus neque mittatis margaritas vestras ante porcos ne forte conculcent eas pedibus suis."
> Cum es damnatorum possederunt porcos iens ut salire off sanguinum cliff in abyssum?"
> --
> View this message in context: http://spssx-discussion.1045642.n5.nabble.com/spss-sYntax-to-get-eigenvalues-tp5723185p5723202.html
> Sent from the SPSSX Discussion mailing list archive at Nabble.com.
>
> =====================
> To manage your subscription to SPSSX-L, send a message to
> [hidden email] (not to SPSSX-L), with no body text except the
> command. To leave the list, send the command
> SIGNOFF SPSSX-L
> For a list of commands to manage subscriptions, send the command
> INFO REFCARD
>
>

[Kirill Orlov]

Kirill,
Please enlighten the class as to how an SVD is NOT the same as an
eigendecomposition.
In my example Q are eigenvalues, U and V are orthonormal .
If one does an SVD of a symmetrical matrix the results are identical to
EIGEN.

Eigenvalue problem is A = V*D*inv(V), where A is square decomposed matrix, D is diagonal matrix with eigenvalues , V is the eigenvectors. Note - there is only one type of eigenvectors.
If A is symmetric, inv(V)=t(V). If A is asymmetric, there emerge difficulties and variants - how and when to compute V. SPSS MATRIX (being so old) unfortunately is missing a function for that.

Singular-value problem is A = U*D*t(V), where A can be of any shape. D is the diagonal matrix with singular values. Note that there are two types of eigenvectors - left U and right V. Only when A is square symmetric positive semidefinite, the decomposition becomes equivalent to eigendecomposition (then, U=V, eigenvalues = singular values): for example, you can do PCA analysis either way.

So, (the missing) EIGEN for a square asymmetric matrix is not the same as SDV of it.

Below is the the code of eigendecomposition by QR algorithm (in its simplest, not practically efficient way).
It is for symmetric matrix A and thus gives the results of CALL EIGEN(A,V,D). It can process asymmetric A as well, but V will be wrong then; as for D (diagonal) it might be correct then (but I'm not sure if always correct: I tried just 2 asymmetric matrices, it were correct).

set mxloops 100.
matrix.

COMP A= {0.2273, -0.0852;
        -0.1930,0.1794}.

comp q_= 1.
loop it= 1 to 100.
-comp q= gsch(a).
-comp r= t(q)*a.
-comp a= r*q.
-comp q_= q_*q.
end loop.
print a /title 'Eigenvalues on the diagonal'.
print q_ /title 'Eigenvectors'.
end matrix.

21.11.2013 1:42, David Marso пишет:

Kirill,
Please enlighten the class as to how an SVD is NOT the same as an
eigendecomposition.
In my example Q are eigenvalues, U and V are orthonormal .
If one does an SVD of a symmetrical matrix the results are identical to
EIGEN.

[Kirill Orlov]

Correction a lapse:
"If A is symmetric, inv(V)=t(V)."  Should read "If A is positive semidefinite, inv(V)=t(V)."

Also to say: if A is asymmetric, its eigenvectors V isn't orthogonal: t(V)*V won't be identity matrix.

[David Marso]

Thanks for the elaboration!
I did a little mod of your code and found something interesting.
First I changed it to stop when the iterations converged rather than 100.
OTOH:  When experimenting with an arbitrary matrix it failed to converge even after 100000 iterations.
Probably something bizarre about the matrix itself?  Thoughts?
--
set mxloops 100000.
MATRIX.
COMPUTE epsilon= 1E-18.
COMPUTE A=
 {0.2273 , -0.0852,.2345,.2345;
 -0.1930 ,0.1794,.24567,.12342;
 .378    ,.2783  ,.7897,.3454;
 .238934 ,.234675,.1234,.4567}.
CALL SVD(A,X,Y,Z).
PRINT A.
PRINT X.
PRINT Y.
PRINT Z.
COMPUTE q_= 1.
COMPUTE solution={ a, a, a}.
LOOP it= 1 to 100000.
+  DO IF (it GT 1).
+    COMPUTE solution={ a, q, q_ }.
+  END IF.
+  COMPUTE q= gsch(a).
+  COMPUTE a=  t(q)*a*q.
+  COMPUTE q_= q_*q.
+  COMPUTE delta=MSUM (SSCP(ABS({ a, q, q_ })-ABS(solution))).

END LOOP IF (delta LT epsilon).

PRINT it .
PRINT DELTA /FORMAT "F20.18".
PRINT a /title 'Eigenvalues on the diagonal'.
PRINT q_ /title 'Eigenvectors'.
END MATRIX.

Kirill Orlov wrote

Kirill,
    Please enlighten the class as to how an SVD is NOT the same as an
    eigendecomposition.
    In my example Q are eigenvalues, U and V are orthonormal .
    If one does an SVD of a symmetrical matrix the results are identical to
    EIGEN.

*
Eigenvalue* problem is A = V*D*inv(V), where A is square decomposed
matrix, D is diagonal matrix with eigenvalues , V is the eigenvectors.
Note - there is only one type of eigenvectors.
If A is symmetric, inv(V)=t(V). If A is /asymmetric/, there emerge
difficulties and variants - how and when to compute V. SPSS MATRIX
(being so old) unfortunately is missing a function for that.

*Singular-value* problem is A = U*D*t(V), where A can be of any shape. D
is the diagonal matrix with singular values. Note that there are two
types of eigenvectors - left U and right V. Only when A is *square
symmetric positive semidefinite*, the decomposition becomes equivalent
to eigendecomposition (then, U=V, eigenvalues = singular values): for
example, you can do PCA analysis either way.

So, (the missing) EIGEN for a square asymmetric matrix is not the same
as SDV of it.

Below is the the code of eigendecomposition by QR algorithm (in its
simplest, not practically efficient way).
It is for symmetric matrix A and thus gives the results of CALL
EIGEN(A,V,D). It can process asymmetric A as well, but V will be *wrong
*then; as for D (diagonal) it *might *be correct then (but I'm /not
/sure if always correct: I tried just 2 asymmetric matrices, it were
correct).

set mxloops 100.
matrix.

COMP A= {0.2273, -0.0852;
         -0.1930,0.1794}.

comp q_= 1.
loop it= 1 to 100.
-comp q= gsch(a).
-comp r= t(q)*a.
-comp a= r*q.
-comp q_= q_*q.
end loop.
print a /title 'Eigenvalues on the diagonal'.
print q_ /title 'Eigenvectors'.
end matrix.


21.11.2013 1:42, David Marso ?????:
> Kirill,
> Please enlighten the class as to how an SVD is NOT the same as an
> eigendecomposition.
> In my example Q are eigenvalues, U and V are orthonormal .
> If one does an SVD of a symmetrical matrix the results are identical to
> EIGEN.
>

[Kirill Orlov]

David, as I've said, this was the code for symmetric matrix, your matrix is not symmetric. Second, not every asymmetric matric can have eigenvectors. e.g. http://en.wikipedia.org/wiki/Defective_matrix

21.11.2013 18:44, David Marso пишет:

Thanks for the elaboration!
I did a little mod of your code and found something interesting.
First I changed it to stop when the iterations converged rather than 100.
OTOH:  When experimenting with an arbitrary matrix it failed to converge
even after 100000 iterations.
Probably something bizarre about the matrix itself?  Thoughts?
--
set mxloops 100000.
MATRIX.
COMPUTE epsilon= 1E-18.
COMPUTE A=
 {0.2273 , -0.0852,.2345,.2345;
 -0.1930 ,0.1794,.24567,.12342;
 .378    ,.2783  ,.7897,.3454;
 .238934 ,.234675,.1234,.4567}.
CALL SVD(A,X,Y,Z).
PRINT A.
PRINT X.
PRINT Y.
PRINT Z.
COMPUTE q_= 1.
COMPUTE solution={ a, a, a}.
LOOP it= 1 to 100000.
+  DO IF (it GT 1).
+    COMPUTE solution={ a, q, q_ }.
+  END IF.
+  COMPUTE q= gsch(a).
+  COMPUTE a=  t(q)*a*q.
+  COMPUTE q_= q_*q.
+  COMPUTE delta=MSUM (SSCP(ABS({ a, q, q_ })-ABS(solution))).

END LOOP IF (delta LT epsilon).

PRINT it .
PRINT DELTA /FORMAT "F20.18".
PRINT a /title 'Eigenvalues on the diagonal'.
PRINT q_ /title 'Eigenvectors'.
END MATRIX.


Kirill Orlov wrote

Kirill,
    Please enlighten the class as to how an SVD is NOT the same as an
    eigendecomposition.
    In my example Q are eigenvalues, U and V are orthonormal .
    If one does an SVD of a symmetrical matrix the results are identical
to
    EIGEN.

*
Eigenvalue* problem is A = V*D*inv(V), where A is square decomposed
matrix, D is diagonal matrix with eigenvalues , V is the eigenvectors.
Note - there is only one type of eigenvectors.
If A is symmetric, inv(V)=t(V). If A is /asymmetric/, there emerge
difficulties and variants - how and when to compute V. SPSS MATRIX
(being so old) unfortunately is missing a function for that.

*Singular-value* problem is A = U*D*t(V), where A can be of any shape. D
is the diagonal matrix with singular values. Note that there are two
types of eigenvectors - left U and right V. Only when A is *square
symmetric positive semidefinite*, the decomposition becomes equivalent
to eigendecomposition (then, U=V, eigenvalues = singular values): for
example, you can do PCA analysis either way.

So, (the missing) EIGEN for a square asymmetric matrix is not the same
as SDV of it.

Below is the the code of eigendecomposition by QR algorithm (in its
simplest, not practically efficient way).
It is for symmetric matrix A and thus gives the results of CALL
EIGEN(A,V,D). It can process asymmetric A as well, but V will be *wrong
*then; as for D (diagonal) it *might *be correct then (but I'm /not
/sure if always correct: I tried just 2 asymmetric matrices, it were
correct).

set mxloops 100.
matrix.

COMP A= {0.2273, -0.0852;
         -0.1930,0.1794}.

comp q_= 1.
loop it= 1 to 100.
-comp q= gsch(a).
-comp r= t(q)*a.
-comp a= r*q.
-comp q_= q_*q.
end loop.
print a /title 'Eigenvalues on the diagonal'.
print q_ /title 'Eigenvectors'.
end matrix.


21.11.2013 1:42, David Marso ?????:

Kirill,
Please enlighten the class as to how an SVD is NOT the same as an
eigendecomposition.
In my example Q are eigenvalues, U and V are orthonormal .
If one does an SVD of a symmetrical matrix the results are identical to
EIGEN.

-----
Please reply to the list and not to my personal email.
Those desiring my consulting or training services please feel free to email me.
---
"Nolite dare sanctum canibus neque mittatis margaritas vestras ante porcos ne forte conculcent eas pedibus suis."
Cum es damnatorum possederunt porcos iens ut salire off sanguinum cliff in abyssum?"
--
View this message in context: http://spssx-discussion.1045642.n5.nabble.com/spss-sYntax-to-get-eigenvalues-tp5723185p5723225.html
Sent from the SPSSX Discussion mailing list archive at Nabble.com.

=====================
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[hidden email] (not to SPSSX-L), with no body text except the
command. To leave the list, send the command
SIGNOFF SPSSX-L
For a list of commands to manage subscriptions, send the command
INFO REFCARD