Inmathematics, thePoincaré separation theorem, also known as theCauchy interlacing theorem, gives someupper and lower bounds ofeigenvalues of a realsymmetric matrixB^TAB that can be considered as theorthogonal projection of a larger real symmetric matrixA onto alinear subspace spanned by the columns of B. The theorem is named afterHenri Poincaré.

More specifically, letA be ann × n real symmetric matrix andB ann × rsemi-orthogonal matrix such thatB^TB =I_r. Denote by${\displaystyle {\lambda }_i}$,i = 1, 2, ..., n and${\displaystyle {\mu }_i}$,i = 1, 2, ..., r the eigenvalues ofA andB^TAB, respectively (in descending order). We have

${\displaystyle {\lambda }_i\ge {\mu }_i\ge {\lambda }_{n-r+i},}$

An algebraic proof, based on thevariational interpretation of eigenvalues, has been published in Magnus'Matrix Differential Calculus with Applications in Statistics and Econometrics.^[1] From the geometric point of view,B^TAB can be considered as theorthogonal projection ofA onto the linear subspace spanned byB, so the above results follow immediately.^[2]

An alternative proof can be made for the case whereB is a principal submatrix ofA, demonstrated by Steve Fisk.^[3]

When considering two mechanical systems, each described by anequation of motion, that differ by exactly one constraint (such that${\displaystyle n-r=1}$), thenatural frequencies of the two systems interlace.

This has an important consequence when considering thefrequency response of a complicated system such as alarge room. Even though there may be many modes, each with unpredictable modes shapes that will vary as details change such as furniture being moved, the interlacing theorem implies that the modal density (average number of modes per frequency interval) remains predictable and approximately constant. This allows for the technique ofmodal density analysis.

Min-max theorem#Cauchy interlacing theorem

• Wolfram Alpha."Poincaré Separation Theorem".

• Inequalities (mathematics)
• Matrix theory
• Theorems in linear algebra

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