|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 Eckart-Young 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
number of dimensions or factors need to account for the data in the
This is the most widely known method of representing both constructs
elements in a single analysis and was originally introduced by Patrick
who confusingly promulgated it initially as ‘The Principal Components
a Repertory Grid’ (Slater, 1964) although it is now more commonly known
as INGRID. The singular-values are the
roots of the eigenvalues of the cross-products matrix. The procedure
two sets of eigenvectors, (see principal components
for a brief definition of eigenvalues and eigenvectors) one for the
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
for the constructs which reflect the importance of the ‘factor’ (in
of variance accounted for), and the column (element) eigenvectors
rescaled. The two loading matrices may be then be multiplied together
produce an approximation of the original matrix (grid) which is an
over principal components and which only reproduces correlations.
advantage is that a singular value decomposition can be carried out
there is no variation in a row or column of the grid.
analysis is usually accompanied by some form of pre-scaling of the grid
to remove statistical artefacts. For example, INGRID removes the effect
(row) construct means to avoid them influencing the derived loadings.
approaches suggest double-centring, or removing both element and
C., and Young, G. (1936) Approximation of one matrix by another of
rank. Psychometrika, 1, 211-218.
P. (1964) The principal components of a repertory grid.
London: Vincent Andrews.
Richard C. Bell