Mixing
On positive definiteness of a sub-matrix after first step Gaussian elimination to a symmetric positive...
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Let $A=[a_{ij}]in M_n(mathbb R)$ be a symmetric positive definite matrix (i.e. all eigenvalues of $A$ are real and positive ) with $a_{11}ne 0$ .
Now let $A_1=[a' _{jk}] in M_{n-1}(mathbb R)$ be defined as $a'_{jk}:=a_{jk}-dfrac{a_{j1}a_{1k}}{a_{11}},forall 2le j,kle n$. Then notice that $A_1$ is symmetric .
My question is : Is $A_1$ positive definite i.e. are all the eigenvalues of $A_1$ positive ?
Note that $A_1$ is the lower block of $A$ after first step of Gaussian elimination to $A$ has been done to make all the first entries of all the 2-nd to $n$-th rows of $A$ zero while retaining first row of $A$ intact.
linear-algebra matrices eigenvalues-eigenvectors positive-definite gaussian-elimination
asked Jan 16 at 10:04
user521337user521337
1,1881416
$endgroup$
add a comment |
$begingroup$
Let $A=[a_{ij}]in M_n(mathbb R)$ be a symmetric positive definite matrix (i.e. all eigenvalues of $A$ are real and positive ) with $a_{11}ne 0$ .
Now let $A_1=[a' _{jk}] in M_{n-1}(mathbb R)$ be defined as $a'_{jk}:=a_{jk}-dfrac{a_{j1}a_{1k}}{a_{11}},forall 2le j,kle n$. Then notice that $A_1$ is symmetric .
My question is : Is $A_1$ positive definite i.e. are all the eigenvalues of $A_1$ positive ?
Note that $A_1$ is the lower block of $A$ after first step of Gaussian elimination to $A$ has been done to make all the first entries of all the 2-nd to $n$-th rows of $A$ zero while retaining first row of $A$ intact.
linear-algebra matrices eigenvalues-eigenvectors positive-definite gaussian-elimination
asked Jan 16 at 10:04
user521337user521337
1,1881416
$endgroup$
add a comment |
$begingroup$
Let $A=[a_{ij}]in M_n(mathbb R)$ be a symmetric positive definite matrix (i.e. all eigenvalues of $A$ are real and positive ) with $a_{11}ne 0$ .
Now let $A_1=[a' _{jk}] in M_{n-1}(mathbb R)$ be defined as $a'_{jk}:=a_{jk}-dfrac{a_{j1}a_{1k}}{a_{11}},forall 2le j,kle n$. Then notice that $A_1$ is symmetric .
My question is : Is $A_1$ positive definite i.e. are all the eigenvalues of $A_1$ positive ?
Note that $A_1$ is the lower block of $A$ after first step of Gaussian elimination to $A$ has been done to make all the first entries of all the 2-nd to $n$-th rows of $A$ zero while retaining first row of $A$ intact.
linear-algebra matrices eigenvalues-eigenvectors positive-definite gaussian-elimination
asked Jan 16 at 10:04
user521337user521337
1,1881416
$endgroup$
Let $A=[a_{ij}]in M_n(mathbb R)$ be a symmetric positive definite matrix (i.e. all eigenvalues of $A$ are real and positive ) with $a_{11}ne 0$ .
Now let $A_1=[a' _{jk}] in M_{n-1}(mathbb R)$ be defined as $a'_{jk}:=a_{jk}-dfrac{a_{j1}a_{1k}}{a_{11}},forall 2le j,kle n$. Then notice that $A_1$ is symmetric .
My question is : Is $A_1$ positive definite i.e. are all the eigenvalues of $A_1$ positive ?
Note that $A_1$ is the lower block of $A$ after first step of Gaussian elimination to $A$ has been done to make all the first entries of all the 2-nd to $n$-th rows of $A$ zero while retaining first row of $A$ intact.
linear-algebra matrices eigenvalues-eigenvectors positive-definite gaussian-elimination
linear-algebra matrices eigenvalues-eigenvectors positive-definite gaussian-elimination
asked Jan 16 at 10:04
user521337user521337
1,1881416
asked Jan 16 at 10:04
user521337user521337
1,1881416
asked Jan 16 at 10:04
user521337user521337
1,1881416
asked Jan 16 at 10:04
user521337user521337
1,1881416
asked Jan 16 at 10:04
user521337user521337
1,1881416
1,1881416
add a comment |
add a comment |
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