Is there any specific relationship among the determinant of leading principal submatrices of a tridiagonal matrix?
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The following symmetric matrix $A$ is given. Let denote $A_i$ the $i$-th leading principal submatrix of the matrix $A$.
$$
A = beginbmatrix
a_1 & b_1 \
b_1 & a_2 & b_2 \
& b_2 & ddots & ddots \
& & ddots & ddots & b_n-1 \
& & & b_n-1 & a_n
endbmatrix
$$
The determinant of the matrix $A$ is given by the recurrence relation
$det(A_i) = a_i det(A_i-1) - b_i-1^2det(A_i-2)$
I would like to know is there any other relation among the determinant of the leading principal submatrix of $A$? for example a numerically that determines determinant of $A_j$ is greater than $A_m$ for $j > m$ or any thing else.
linear-algebra matrices determinant tridiagonal-matrices
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up vote
2
down vote
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The following symmetric matrix $A$ is given. Let denote $A_i$ the $i$-th leading principal submatrix of the matrix $A$.
$$
A = beginbmatrix
a_1 & b_1 \
b_1 & a_2 & b_2 \
& b_2 & ddots & ddots \
& & ddots & ddots & b_n-1 \
& & & b_n-1 & a_n
endbmatrix
$$
The determinant of the matrix $A$ is given by the recurrence relation
$det(A_i) = a_i det(A_i-1) - b_i-1^2det(A_i-2)$
I would like to know is there any other relation among the determinant of the leading principal submatrix of $A$? for example a numerically that determines determinant of $A_j$ is greater than $A_m$ for $j > m$ or any thing else.
linear-algebra matrices determinant tridiagonal-matrices
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
The following symmetric matrix $A$ is given. Let denote $A_i$ the $i$-th leading principal submatrix of the matrix $A$.
$$
A = beginbmatrix
a_1 & b_1 \
b_1 & a_2 & b_2 \
& b_2 & ddots & ddots \
& & ddots & ddots & b_n-1 \
& & & b_n-1 & a_n
endbmatrix
$$
The determinant of the matrix $A$ is given by the recurrence relation
$det(A_i) = a_i det(A_i-1) - b_i-1^2det(A_i-2)$
I would like to know is there any other relation among the determinant of the leading principal submatrix of $A$? for example a numerically that determines determinant of $A_j$ is greater than $A_m$ for $j > m$ or any thing else.
linear-algebra matrices determinant tridiagonal-matrices
The following symmetric matrix $A$ is given. Let denote $A_i$ the $i$-th leading principal submatrix of the matrix $A$.
$$
A = beginbmatrix
a_1 & b_1 \
b_1 & a_2 & b_2 \
& b_2 & ddots & ddots \
& & ddots & ddots & b_n-1 \
& & & b_n-1 & a_n
endbmatrix
$$
The determinant of the matrix $A$ is given by the recurrence relation
$det(A_i) = a_i det(A_i-1) - b_i-1^2det(A_i-2)$
I would like to know is there any other relation among the determinant of the leading principal submatrix of $A$? for example a numerically that determines determinant of $A_j$ is greater than $A_m$ for $j > m$ or any thing else.
linear-algebra matrices determinant tridiagonal-matrices
edited Aug 8 at 18:32
Bernard
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110k635103
asked Aug 8 at 18:11
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