非対称特異行列のクリティカルな左固有ベクトルの力学的意味付けとその例説

概要

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In mathematical problems and mechanical engineering, there are a number of examples, in which a non-symmetric Jacobian matrix is involved in solution of simultaneous linear equations. The left and right eigenvectors of such non-symmetric matrices are in general complex and must be well discerned to each other. Their properties and practical meaning are, however, hardly discussed in engineering applications. Especially when the non-symmetric matrix is singular, the critical left eigenvector corresponding to null eigenvalues is of increased significance in the examination of the solvability of the problem. The present paper describes and interprets the substantial role of the critical left eigenvector of the non-symmetric singular matrix in mechanics. Model examples in applied mathematics, solids, structures and rigid bodies will illustrate the meaning of the critical left eigenvector, when the singularity is unavoidable in the problem to be solved. The discussion will be then extended to the critical left singular vector of a rectangular matrix.
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