Mixing
On $QR$ factorization of upper Hessenberg matrices with real entries
$begingroup$
Let $A$ be an $ntimes n$ invertible upper Hessenberg matrix (https://en.wikipedia.org/wiki/Hessenberg_matrix) with real entries. I am trying to prove the following three statements :
(1) There exists decomposition $A=QR$ where $Q$ is orthogonal and $R$ is upper triangular and $Q=H_1...H_{n-1}$ for some Householder matrices $H_1,...,H_{n-1}$. (https://en.wikipedia.org/wiki/Householder_transformation )
(2) $Q$ as in (1) is upper Hessenberg
(3) $RQ$, where $Q,R$ are as in (1), is upper Hessenberg.
Any proof or any reference of the statements with proof will be greatly appreciated.
Thanks in advance
linear-algebra matrices
asked Jan 16 at 12:49
user521337user521337
1,1881416
$endgroup$
add a comment |
$begingroup$
Let $A$ be an $ntimes n$ invertible upper Hessenberg matrix (https://en.wikipedia.org/wiki/Hessenberg_matrix) with real entries. I am trying to prove the following three statements :
(1) There exists decomposition $A=QR$ where $Q$ is orthogonal and $R$ is upper triangular and $Q=H_1...H_{n-1}$ for some Householder matrices $H_1,...,H_{n-1}$. (https://en.wikipedia.org/wiki/Householder_transformation )
(2) $Q$ as in (1) is upper Hessenberg
(3) $RQ$, where $Q,R$ are as in (1), is upper Hessenberg.
Any proof or any reference of the statements with proof will be greatly appreciated.
Thanks in advance
linear-algebra matrices
asked Jan 16 at 12:49
user521337user521337
1,1881416
$endgroup$
add a comment |
$begingroup$
Let $A$ be an $ntimes n$ invertible upper Hessenberg matrix (https://en.wikipedia.org/wiki/Hessenberg_matrix) with real entries. I am trying to prove the following three statements :
(1) There exists decomposition $A=QR$ where $Q$ is orthogonal and $R$ is upper triangular and $Q=H_1...H_{n-1}$ for some Householder matrices $H_1,...,H_{n-1}$. (https://en.wikipedia.org/wiki/Householder_transformation )
(2) $Q$ as in (1) is upper Hessenberg
(3) $RQ$, where $Q,R$ are as in (1), is upper Hessenberg.
Any proof or any reference of the statements with proof will be greatly appreciated.
Thanks in advance
linear-algebra matrices
asked Jan 16 at 12:49
user521337user521337
1,1881416
$endgroup$
Let $A$ be an $ntimes n$ invertible upper Hessenberg matrix (https://en.wikipedia.org/wiki/Hessenberg_matrix) with real entries. I am trying to prove the following three statements :
(1) There exists decomposition $A=QR$ where $Q$ is orthogonal and $R$ is upper triangular and $Q=H_1...H_{n-1}$ for some Householder matrices $H_1,...,H_{n-1}$. (https://en.wikipedia.org/wiki/Householder_transformation )
(2) $Q$ as in (1) is upper Hessenberg
(3) $RQ$, where $Q,R$ are as in (1), is upper Hessenberg.
Any proof or any reference of the statements with proof will be greatly appreciated.
Thanks in advance
linear-algebra matrices
linear-algebra matrices
asked Jan 16 at 12:49
user521337user521337
1,1881416
asked Jan 16 at 12:49
user521337user521337
1,1881416
asked Jan 16 at 12:49
user521337user521337
1,1881416
asked Jan 16 at 12:49
user521337user521337
1,1881416
asked Jan 16 at 12:49
user521337user521337
1,1881416
1,1881416
add a comment |
add a comment |
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