Mixing
Modification of Poincaré Separation Theorem
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We know from Poincaré's separation theorem that for semi-orthogonal $mathbf{B} in mathbb{R}^{ntimes k}$ and real, symmetric $mathbf{A} in mathbb{R}^{n times n}$ with eigenvalues $lambda_1 > lambda_2 > ... > lambda_n$, the product $mathbf{B}^T mathbf{AB}$ has eigenvalues $mu_i$ such that,
$$ lambda_i geq mu_i geq lambda_{n-k+i}, quad i = 1,2,...,k$$
From searching online there seems to be a proof for this theorem in Magnus and Neudecker, "Matrix differential calculus with applications in statistics and econometrics," but I do not access to the text. I'm sure this proof would shed light on my following question.
My question: is there any way to find a similar result for the product $mathbf{ABB}^T$, i.e. can we bound the eigenvalues of the product relative to the eigenvalues of the matrix $mathbf{A}$?
linear-algebra eigenvalues-eigenvectors
edited Jan 10 at 22:00
Bernard
120k740115
asked Jan 10 at 21:54
SemiPolishSemiPolish
12
$endgroup$
add a comment |
$begingroup$
We know from Poincaré's separation theorem that for semi-orthogonal $mathbf{B} in mathbb{R}^{ntimes k}$ and real, symmetric $mathbf{A} in mathbb{R}^{n times n}$ with eigenvalues $lambda_1 > lambda_2 > ... > lambda_n$, the product $mathbf{B}^T mathbf{AB}$ has eigenvalues $mu_i$ such that,
$$ lambda_i geq mu_i geq lambda_{n-k+i}, quad i = 1,2,...,k$$
From searching online there seems to be a proof for this theorem in Magnus and Neudecker, "Matrix differential calculus with applications in statistics and econometrics," but I do not access to the text. I'm sure this proof would shed light on my following question.
My question: is there any way to find a similar result for the product $mathbf{ABB}^T$, i.e. can we bound the eigenvalues of the product relative to the eigenvalues of the matrix $mathbf{A}$?
linear-algebra eigenvalues-eigenvectors
edited Jan 10 at 22:00
Bernard
120k740115
asked Jan 10 at 21:54
SemiPolishSemiPolish
12
$endgroup$
add a comment |
$begingroup$
We know from Poincaré's separation theorem that for semi-orthogonal $mathbf{B} in mathbb{R}^{ntimes k}$ and real, symmetric $mathbf{A} in mathbb{R}^{n times n}$ with eigenvalues $lambda_1 > lambda_2 > ... > lambda_n$, the product $mathbf{B}^T mathbf{AB}$ has eigenvalues $mu_i$ such that,
$$ lambda_i geq mu_i geq lambda_{n-k+i}, quad i = 1,2,...,k$$
From searching online there seems to be a proof for this theorem in Magnus and Neudecker, "Matrix differential calculus with applications in statistics and econometrics," but I do not access to the text. I'm sure this proof would shed light on my following question.
My question: is there any way to find a similar result for the product $mathbf{ABB}^T$, i.e. can we bound the eigenvalues of the product relative to the eigenvalues of the matrix $mathbf{A}$?
linear-algebra eigenvalues-eigenvectors
edited Jan 10 at 22:00
Bernard
120k740115
asked Jan 10 at 21:54
SemiPolishSemiPolish
12
$endgroup$
We know from Poincaré's separation theorem that for semi-orthogonal $mathbf{B} in mathbb{R}^{ntimes k}$ and real, symmetric $mathbf{A} in mathbb{R}^{n times n}$ with eigenvalues $lambda_1 > lambda_2 > ... > lambda_n$, the product $mathbf{B}^T mathbf{AB}$ has eigenvalues $mu_i$ such that,
$$ lambda_i geq mu_i geq lambda_{n-k+i}, quad i = 1,2,...,k$$
From searching online there seems to be a proof for this theorem in Magnus and Neudecker, "Matrix differential calculus with applications in statistics and econometrics," but I do not access to the text. I'm sure this proof would shed light on my following question.
My question: is there any way to find a similar result for the product $mathbf{ABB}^T$, i.e. can we bound the eigenvalues of the product relative to the eigenvalues of the matrix $mathbf{A}$?
linear-algebra eigenvalues-eigenvectors
linear-algebra eigenvalues-eigenvectors
edited Jan 10 at 22:00
Bernard
120k740115
asked Jan 10 at 21:54
SemiPolishSemiPolish
12
edited Jan 10 at 22:00
Bernard
120k740115
asked Jan 10 at 21:54
SemiPolishSemiPolish
12
edited Jan 10 at 22:00
Bernard
120k740115
edited Jan 10 at 22:00
Bernard
120k740115
edited Jan 10 at 22:00
Bernard
120k740115
120k740115
asked Jan 10 at 21:54
SemiPolishSemiPolish
12
asked Jan 10 at 21:54
SemiPolishSemiPolish
12
asked Jan 10 at 21:54
SemiPolishSemiPolish
12
12
add a comment |
add a comment |
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