
Singularvaluedecomposition

This technique is allied to both principal components and correspondence
analysis. It was originally devised by Eckart and Young in 1936
(and is sometimes known as EckartYoung decomposition) who showed that
a matrix (such as a repertory grid)
can be approximated
by another matrix of lower rank (rank being a property of matrices akin
to
number of dimensions or factors need to account for the data in the
matrix).
This is the most widely known method of representing both constructs
and
elements in a single analysis and was originally introduced by Patrick
Slater
who confusingly promulgated it initially as ‘The Principal Components
of
a Repertory Grid’ (Slater, 1964) although it is now more commonly known
as INGRID. The singularvalues are the
square
roots of the eigenvalues of the crossproducts matrix. The procedure
produces
two sets of eigenvectors, (see principal components
for a brief definition of eigenvalues and eigenvectors) one for the
rows
of the matrix and one for the columns. The row (construct) eigenvectors
can be multiplied by the square root of the singular values to give
loadings
for the constructs which reflect the importance of the ‘factor’ (in
terms
of variance accounted for), and the column (element) eigenvectors
similarly
rescaled. The two loading matrices may be then be multiplied together
to
produce an approximation of the original matrix (grid) which is an
advantage
over principal components and which only reproduces correlations.
Another
advantage is that a singular value decomposition can be carried out
when
there is no variation in a row or column of the grid.
Singularvaluedecomposition
analysis is usually accompanied by some form of prescaling of the grid
to remove statistical artefacts. For example, INGRID removes the effect
of
(row) construct means to avoid them influencing the derived loadings.
Other
approaches suggest doublecentring, or removing both element and
construct
means. 
References

 Eckart,
C., and Young, G. (1936) Approximation of one matrix by another of
lower
rank. Psychometrika, 1, 211218.
 Slater,
P. (1964) The principal components of a repertory grid.
London: Vincent Andrews.

Richard C. Bell

