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Matrix representation of ideas: stimulating creativity using matrix Algebra

Victor Tang (Massachusetts Institute of Technology, Cambridge, Massachusetts, USA)

International Journal of Innovation Science

ISSN: 1757-2223

Article publication date: 29 October 2019

Issue publication date: 21 November 2019




The purpose of this paper is to present a fresh approach to stimulate individual creativity. It introduces a mathematical representation for creative ideas, six creativity operators and methods of matrix-algebra to evaluate, improve and stimulate creative ideas. Creativity begins with ideas to resolve a problem or tackle an opportunity. By definition, a creative idea must be simultaneously novel and useful. To inject analytic rigor into these concepts of creative ideas, the author introduces a feature-attribute matrix-construct to represent ideas, creativity operators that use ideas as operands and methods of matrix algebra. It is demonstrated that it is now possible to analytically and quantitatively evaluate the intensity of the variables that make an idea more, equal or less, creative than another. The six creativity operators are illustrated with detailed multi-disciplinary real-world examples. The mathematics and working principles of each creativity operator are discussed.


The unit of analysis is ideas, not theory. Ideas are man-made artifacts. They are represented by an original feature-attribute matrix construct. Using matrix algebra, idea matrices can be manipulated to improve their creative intensity, which are now quantitatively measurable. Unlike atoms and cute rabbits, creative ideas, do not occur in nature. Only people can conceive and develop creative ideas for embodiment in physical, non-physical forms, or in a mix of both. For example, as widgets, abstract theorems, business processes, symphonies, organization structures, and so on. The feature-attribute matrix construct is used to represent novelty and usefulness. The multiplicative product of these two matrices forms the creativity matrix. Six creativity operators and matrix algebra are introduced to stimulate and measure creative ideas. Creativity operators use idea matrices as operands. Uses of the six operators are demonstrated using multi-disciplinary real-world examples. Metrics for novelty, usefulness and creativity are in ratio scales, grounded on the Weber–Fechner Law. This law is about persons’ ability to discern differences in the intensity of stimuli.


Ideas are represented using feature-attribute matrices. This construct is used to represent novel, useful and creative ideas with more clarity and precision than before. Using matrices, it is shown how to unambiguously and clearly represent creative ideas endowed with novelty and usefulness. It is shown that using matrix algebra, on idea matrices, makes it possible to analyze multi-disciplinary, real-world cases of creative ideas, with clarity and discriminatory power, to uncover insights about novelty and usefulness. Idea-matrices and the methods of matrix algebra have strong explanatory and predictive power. Using of matrix algebra and eigenvalue analyses, of idea-matrices, it is demonstrated how to quantitatively rank ideas, features and attributes of creative ideas. Matrix methods operationalize and quantitatively measure creativity, novelty and usefulness. The specific elementary variables that characterize creativity, novelty and usefulness factors, can now be quantitatively ranked. Creativity, novelty and usefulness factors are not considered as monolithic, irreducible factors, vague “lumpy” qualitative factors, but as explicit sets of elementary, specific and measurable variables in ratio scales. This significantly improves the acuity and discriminatory power in the analyses of creative ideas. The feature-attribute matrix approach and its matrix operators are conceptually consistent and complementary with key extant theories engineering design and creativity.


First to define and specify ideas as feature-attribute matrices. It is demonstrated that creative ideas, novel ideas and useful ideas can be analytically and unambiguously specified and measured for creativity. It is significant that verbose qualitative narratives will no longer be the exclusive means to specify creative ideas. Rather, qualitative narratives will be used to complement the matrix specifications of creative ideas. First to specify six creativity operators enabling matrix algebra to operate on idea-matrices as operands to generate new ideas. This capability informs and guides a person’s intuition. The myth and dependency, on non-repeatable or non-reproducible serendipity, flashes of “eureka” moments or divine inspiration, can now be vacated. Though their existence cannot be ruled out. First to specify matrix algebra and eigen-value methods of quantitative analyses of feature-attribute matrices to rank the importance of elementary variables that characterize factors of novelty, usefulness and creativity. Use of verbose qualitative narratives of novelty, usefulness and creativity as monolithic “lumpy” factors can now be vacated. Such lumpy narratives risk being ambiguous, imprecise, unreliable and non-reproducible, Analytic and quantitative methods are more reliable and consistent. First to define and specify a method of “attacking the negatives” to systematically pinpoint the improvements of an idea’s novelty, usefulness and creativity. This procedure informs and methodically guides the improvements of deficient ideas.



The author thanks J. Nadan for his encouragement especially when the concepts were nascent; reviewers for their constructive comments; Y. Reich for pointing us to key publications; J.X. Luo for the laptop-covers example and selected Section 4.3 calculations; and discussions with F. Yanine. This article is based on ideas presented at the Innova-Con 2014 Conference in New York City.


Tang, V. (2018), "Matrix representation of ideas: stimulating creativity using matrix Algebra", International Journal of Innovation Science, Vol. 11 No. 4, pp. 489-538.



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Copyright © 2019, Emerald Publishing Limited

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