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The matrices of factor analysis. (English) Zbl 0063.00040

From the text: In factor analysis a study is made of \(n\)-rowed symmetric matrices \(M_0\) whose diagonal elements are all zero and whose nondiagonal elements are correlation coefficients, that is, real numbers between zero and unity. A general problem is then to determine a diagonal matrix \(D\) such that \(M=M_0+D\) has minimum rank \(p\). The diagonal elements of \(D\) are those of the factor matrix \(M\) and are called the communalities. As soon as \(D\) is evaluated a standard computational technique may be employed to write \(M\) as a product \(M= FF'\), where \(F\) has \(n\) rows and \(p\) columns, \(F'\) is its transpose.
Professor L. L. Thurstone of the Department of Psychology of the University of Chicago has called my attention to this problem with particular emphasis on a special case. This case is of the greatest importance because the matrices which arise from actual psychological tests appear to be of the type assumed for this case. Our special problem is the study of the case where \[ is equal to what we shall call the ideal rank \(r\). \]

MSC:

62H25 Factor analysis and principal components; correspondence analysis
15A99 Basic linear algebra
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