In
Section 4.6 we saw that the
singular_value_decomposition algorithm in SymPy does things a little bit differently than in
Section 4.6. If we start with a square matrix
\(A\text{,}\) the results are the same, but if
\(A\) is not square, the decomposition
\(A = P\Sigma_A Q^T\) looks a little different. In particular, if
\(A\) is
\(m\times n\text{,}\) the matrix
\(\Sigma_A\) defined in
Section 4.6 will also be
\(m\times n\text{,}\) but it will contain some rows or columns of zeros that are added to get the desired size. The matrix
\(Q\) is an orthogonal
\(n\times n\) matrix whose columns are an orthonormal basis of eigenvectors for
\(A^TA\text{.}\) The matrix
\(P\) is an orthogonal
\(m\times m\) matrix whose columns are an orthonormal basis of
\(\R^m\text{.}\) (The first
\(r\) columns of
\(P\) are given by
\(A\vecq_i\text{,}\) where
\(\vecq_i\) is the eigenvector of
\(A^TA\) corresponding to the positive singular value
\(\sigma_i\text{.}\))
The algorithm provided by SymPy replaces \(\Sigma_A\) by the \(r\times r\) diagonal matrix of nonzero singular values. (This is common in most algorithms, since we don’t want to bother storing data we don’t need.) The matrix \(Q\) is replaced by the \(n\times r\) matrix whose columns are the first \(r\) eigenvectors of \(A^TA\text{,}\) and the matrix \(P\) is replaced by the \(m\times r\) matrix whose columns are the orthonormal basis for the column space of \(A\text{.}\) (Note that the rank of \(A^TA\) is equal to the rank of \(A\text{,}\) which is equal to the number \(r\) of nonzero eigenvectors of \(A^TA\) (counted with multiplicity).)
The product \(P\Sigma_A Q^T\) will be the same in both cases, and the matrix \(P\) is the same as well.
This time, rather than using the SymPy algorithm, we will work through the process outlined in
Section 4.6 step-by-step. Let’s revisit
Example 4.6.4. Let
\(A = \bbm 1\amp 1\amp 1\\1\amp 0\amp -1\ebm\text{.}\) First, we get the singular values:
Next, we get the eigenvalues and eigenvectors of \(A^TA\text{:}\)
Now we need to normalize the eigenvectors, in the correct order. Note that the eigenvectors were listed in increasing order of eigenvalue, so we need to reverse the order. Note that L1 is a list of lists. The eigenvector is the third entry (index 2) in the list (eigenvalue, multiplicity, eigenvector). We also need to turn list elements into matrices. So, for example the second eigenvector is Matrix(L1[1][2]).
Next, we can assemble these vectors into a matrix, and confirm that it’s orthogonal.
We’ve made the matrix \(Q\text{!}\) Next, we construct \(\Sigma_A\text{.}\) This we will do by hand.
Alternatively, you could do SigA = diag(L0[0],L0[1]).row_join(Matrix([0,0])). Finally, we need to make the matrix \(P\text{.}\) First, we find the vectors \(A\vecq_1, A\vecq_2\) and normalize. (Note that \(A\vecq_3=\zer\text{,}\) so this vector is unneeded, as expected.)
Note that the matrix \(P\) is already the correct size, because \(\rank(A)=2\dim(\R^2)\text{.}\) In general, for an \(m\times n\) matrix \(A\text{,}\) if \(\rank(A)=r\lt m\text{,}\) we would have to extend the set \(\{\vecp_1,\ldots, \vecp_r\}\) to a basis for \(\R^m\text{.}\) Finally, we check that our matrices work as advertised.
For convenience, here is all of the above code, with all print commands (except the last one) removed.
from sympy import Matrix,BlockMatrix,init_printing
init_printing()
A = Matrix([[1,1,1],[1,0,-1]])
B=(A.T)*A
L0=A.singular_values()
L1=B.eigenvects()
R1=Matrix(L1[2][2])
R2=Matrix(L1[1][2])
R3=Matrix(L1[0][2])
Q1 = (1/R1.norm())*R1
Q2 = (1/R2.norm())*R2
Q3 = (1/R3.norm())*R3
Q = Matrix(BlockMatrix([Q1,Q2,Q3]))
SigA = diag(L0[0],L0[1]).row_join(Matrix([0,0]))
S1 = A*Q1
S2 = A*Q2
P1 = (1/S1.norm())*S1
P2 = (1/S2.norm())*S2
P = Matrix(BlockMatrix([P1,P2]))
P,SigA,Q,P*SigA*Q.T
