Difference between Singular Value Decomposition and Smith Decomposition

Is there any difference between Singular Value Decomposition and Smith Decomposition?

Is there any relation between the two or they are one and the same thing?

I know Smith decomposition can be applied to only square matrices (in Matlab) and SVD is applicable to any matrix.

However if we take any mxn matrix and derive the SVD and Smith forms, then would they be the same or different?

[Christine Tobler]

The confusion may come from the fact that there is also a Schmidt decomposition, which is essentially the same as the SVD. This appears to be completely independent of the Smith normal form, which has no direct connection to the SVD.

[Paul]

@Christine Tobler

Any insight as to why smithForm() is limited to square matrices? I thought the the Smith form is applicable to non-square matrices as well, at least for the matrix polynomial case.

[Christine Tobler]

I asked a colleague in the Symbolic Toolbox, and the reason here is practical: The algorithm we use only works for square matrices, but is far more efficient than algorithms that would work for any rectangular matrix.

Also, we haven't received any requests for smithForm on rectangular matrices, to my knowledge.

@Siddhanth Sunil Shah, please let us know if you have a need for computing the Smith form on rectangular matrices. For now, my assumption would be you were only exploring the differences with the SVD?

[Paul]

One application of the Smith form for non-square polynomial matrices is to represent a multi-input, multi-output transfer function matrix in Smith-McMillan form. This comment shows a simple example for a square transfer function matrix.

[Siddhanth Sunil Shah]

@Christine Tobler yes i was working on the MIMO transfer function matrices in the Smith form with rectangular matrices but could not proceed ahead with the SmithForm() function available in MATLAB only for square matrices.

I do have a need for computing Smith form for rectangular matrices and wrote an algorithm myself but I guess it does'nt work the way it should.

[Christine Tobler]

Thank you both for the explanations, I have passed this information on to colleagues in the Symbolic Toolbox.

Answer 1

[John D'Errico]

How does this apply to MATLAB? This really is a question purely about linear algebra.

Are they the same? No, they are not. Yes, they look somewhat alike in what they return, ALMOST.

The smithForm applies ONLY to square matrices. But also, it applies ONLY to integer valued matrices. READ THE HELP!

help smithForm

 SMITHFORM   Smith normal form.
    S = SMITHFORM(A) computes the Smith normal form of the square
    invertible matrix A. The elements of A must be integers or 
    univariate polynomials in the variable x = symvar(A,1).
 
    The Smith form S is a diagonal matrix. The first diagonal
    element divides the second, the second divides the third,
    and so on.
 
    [U,V,S] = SMITHFORM(A) also computes unimodular transformation 
    matrices U and V, such that S = U*A*V. 
 
    S = SMITHFORM(A,x) computes the Smith Normal Form of the square
    invertible matrix A. The elements of A are regarded as univariate
    polynomials in the specified variable x.
 
    [U,V,S] = SMITHFORM(A,x) also computes unimodular transformation 
    matrices U and V, such that S = U*A*V. 
 
    Example:
       >> A = sym(invhilb(5));
       >> S = smithForm(A)
 
       S =
           [ 5,  0,   0,   0,    0]
           [ 0, 60,   0,   0,    0]
           [ 0,  0, 420,   0,    0]
           [ 0,  0,   0, 840,    0]
           [ 0,  0,   0,   0, 2520]
 
       >> syms x y 
       >> A = [2/x+y, x^2-y^2; 3*sin(x)+y, x];
       >> [U,V,S] = smithForm(A,y)
 
       U =
           [ 0,       1]
           [ x, y^2-x^2]
 
       V =
           [   0,                 1]
           [ 1/x, -(3*sin(x))/x-y/x]
 
       S =
           [ 1,                                          0]
           [ 0, 3*y^2*sin(x)-3*x^2*sin(x)+y^3+y*(-x^2+x)+2]
 
       >> simplify(U*A*V - S)
 
        ans =
           [ 0, 0]
           [ 0, 0]
 
       >> A = [2*(x-y),3*(x^2-y^2);4*(x^3-y^3),5*(x^4-y^4)];
       >> [U,V,S] = smithForm(A,x)
 
       U =
           [ 0,                     1]
           [ 1, -x/(10*y^3)-3/(5*y^2)]
           
       V =
           [ -x/(4*y^3), -(5*x*y^2)/2-(5*x^2*y)/2-(5*x^3)/2-(5*y^3)/2]
           [  1/(5*y^3),                            2*x^2+2*x*y+2*y^2]
  
       S =
           [ x-y,                       0]
           [   0, x^4+6*x^3*y-6*x*y^3-y^4]
   
    See also HERMITEFORM, JORDAN.

    Documentation for smithForm
       doc smithForm

Next, try it out. Don't know if they are different? TRY IT.

A = magic(3)

A = 3×3

8 1 6 3 5 7 4 9 2

[U,S,V] = svd(A)

U = 3×3

-0.5774 0.7071 0.4082 -0.5774 0.0000 -0.8165 -0.5774 -0.7071 0.4082

S = 3×3

15.0000 0 0 0 6.9282 0 0 0 3.4641

V = 3×3

-0.5774 0.4082 0.7071 -0.5774 -0.8165 -0.0000 -0.5774 0.4082 -0.7071

[U2,S2,V2] = smithForm(A)

U2 = 3×3

0 0 1 0 1 23 1 -164 -3749

S2 = 3×3

0 24 53 1 -10 -22 -4 -3 -7

V2 = 3×3

1 0 0 0 1 0 0 0 360

They don't look the same to me. But then I do need a new set of glasses.

https://en.wikipedia.org/wiki/Smith_normal_form

Finally, if you try using smithform even on a square matrix that has non-integer elements, it will fail.

A = randn(2)

A = 2×2

-1.4674 -0.1167 -0.3505 -0.0260

svd(A)

ans = 2×1

1.5134 0.0018

smithForm(A)

Error using mupadengine/feval_internal
Unable to convert the matrix entries to integers or univariate polynomials.

Error in smithForm (line 96)
Usym = feval_internal(symengine,'symobj::smithForm',A);

The Smith normal form is designed to solve a totally different set of problems compared to when one would use the svd. They are different tools, with different properties, different purposes, and different results.

[Paul]

As stated in the Help, the function smithForm() also applies to symbolic, polynmomial matrices, not just integer valued matrices.

While true that the smithForm() function is limited to square matrices, it's worth pointing out that that is a limitation of Matlab's implementation and not of the Smith form in general, as shown in the wikipedia link.

As shown in the Help, this line should be

%[U2,S2,V2] = smithForm(A)
[U2,V2,S2] = smithForm(A)

which, of course, wouldn't change the conclusion that Smith and SVD are different.

As an aside, I'm not too surprised that smithForm() accepts a double as input, but I am surprised it returns double (not sym) as output.

[John D'Errico]

Agreed, as part of the symbolic TB, it would make sense for it to return a symbolic result.

I do wish they had chosen different names for the return variables. While it is completely irrelevant what they call the variables, people will see U,S,V, and get confused, thinking the two are the same.

Answer 2

[Siddhanth Sunil Shah]

Thank you for your insights! :)