Elimination Matrix Specific entry question The Next CEO of Stack OverflowHow is row...
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Elimination Matrix Specific entry question
The Next CEO of Stack OverflowHow is row elimination getting rid of this entry?Rotation Matrix inverse using Gauss-Jordan eliminationMirror Matrix MultiplicationSimple Eigenvalue finding question (by gauss elimination)Premultiplication by an elimination matrixHow to determine the transition matrices when doing Gaussian elimination?How to find the $LU$ factorization of a matrix $A$ when elimination breaks downGuassian Elimination with matricesOrthogonal matrix with single $0$ entryElementary Matrix Multiplication and Gauss Elimination
$begingroup$
Given a $3 times 3$ array $$
A =begin{bmatrix} 1 & 2 &1 \ 3 & 8 & 1 \ 0 &4 &1 end{bmatrix},
$$
My understanding is that that I subtract $3$ times row one from row two to eliminate entry $a_{21}$. Therefore $E_{21}$ should be
$$
E_{21} = begin{bmatrix} 1 & 1 & 1 \ mathbf{0} & 2 & -2 \ 0 & 4 & 1end{bmatrix}
$$
I see in my notes that elimination by multiplication yields $$
E_{21} = begin{bmatrix} 1 & 0 & 0 \ mathbf{-3} & 1 & 1 \ 0 & 0 & 1end{bmatrix}
$$.
I would like to understand the intuition behind the entries in the multiplication matrix. I am able to follow how one would find the entries for $E_{21}$ using the subtraction method (I highlighted $0$ to illustrate that I understand where that comes from, but I am supposed to use the multiplication method and I do not understand how one reaches these entries. Specifically the highlighted $-3$. Can someone please explain to me how to use the multiplication method to eliminate entries in a matrix?
matrices gaussian-elimination
edited Mar 17 at 16:35
Brian
1,208116
asked Mar 17 at 8:19
AdamAdam
345
$endgroup$
add a comment |
$begingroup$
Given a $3 times 3$ array $$
A =begin{bmatrix} 1 & 2 &1 \ 3 & 8 & 1 \ 0 &4 &1 end{bmatrix},
$$
My understanding is that that I subtract $3$ times row one from row two to eliminate entry $a_{21}$. Therefore $E_{21}$ should be
$$
E_{21} = begin{bmatrix} 1 & 1 & 1 \ mathbf{0} & 2 & -2 \ 0 & 4 & 1end{bmatrix}
$$
I see in my notes that elimination by multiplication yields $$
E_{21} = begin{bmatrix} 1 & 0 & 0 \ mathbf{-3} & 1 & 1 \ 0 & 0 & 1end{bmatrix}
$$.
I would like to understand the intuition behind the entries in the multiplication matrix. I am able to follow how one would find the entries for $E_{21}$ using the subtraction method (I highlighted $0$ to illustrate that I understand where that comes from, but I am supposed to use the multiplication method and I do not understand how one reaches these entries. Specifically the highlighted $-3$. Can someone please explain to me how to use the multiplication method to eliminate entries in a matrix?
matrices gaussian-elimination
edited Mar 17 at 16:35
Brian
1,208116
asked Mar 17 at 8:19
AdamAdam
345
$endgroup$
1
$begingroup$
If I understand you question correctly, you may find this article helpful. Would you mind explaining what a "multiplication matrix" is, and what the "subtraction" and "multiplications" methods to which you refer are? Is each $E_{ij}$ supposed to be an elimination matrix, or a row-equivalent matrix produced at some point during elimination? Your work seems to suggest the latter, while your notes suggest the former (both with some errors).
$endgroup$
– Brian
Mar 17 at 16:32
$begingroup$
You will also note that this very example appear in the article I linked in my previous comment.
$endgroup$
– Brian
Mar 17 at 16:35
$begingroup$
Of course, I meant to say that I am looking to find the left operand multiplied by original Matrix A that is equal to the elimination matrix E(2,1). And then, I am to eliminate entry (3,2)--which should be a "4" in my elimination matrix (2,1). I am struggling with this second step of elimination by matrix multiplication where one needs to fill in the entries for that left hand operand when setting the systems equal to the elimination matrix
$endgroup$
– Adam
Mar 17 at 17:45
add a comment |
$begingroup$
Given a $3 times 3$ array $$
A =begin{bmatrix} 1 & 2 &1 \ 3 & 8 & 1 \ 0 &4 &1 end{bmatrix},
$$
My understanding is that that I subtract $3$ times row one from row two to eliminate entry $a_{21}$. Therefore $E_{21}$ should be
$$
E_{21} = begin{bmatrix} 1 & 1 & 1 \ mathbf{0} & 2 & -2 \ 0 & 4 & 1end{bmatrix}
$$
I see in my notes that elimination by multiplication yields $$
E_{21} = begin{bmatrix} 1 & 0 & 0 \ mathbf{-3} & 1 & 1 \ 0 & 0 & 1end{bmatrix}
$$.
I would like to understand the intuition behind the entries in the multiplication matrix. I am able to follow how one would find the entries for $E_{21}$ using the subtraction method (I highlighted $0$ to illustrate that I understand where that comes from, but I am supposed to use the multiplication method and I do not understand how one reaches these entries. Specifically the highlighted $-3$. Can someone please explain to me how to use the multiplication method to eliminate entries in a matrix?
matrices gaussian-elimination
edited Mar 17 at 16:35
Brian
1,208116
asked Mar 17 at 8:19
AdamAdam
345
$endgroup$
Given a $3 times 3$ array $$
A =begin{bmatrix} 1 & 2 &1 \ 3 & 8 & 1 \ 0 &4 &1 end{bmatrix},
$$
My understanding is that that I subtract $3$ times row one from row two to eliminate entry $a_{21}$. Therefore $E_{21}$ should be
$$
E_{21} = begin{bmatrix} 1 & 1 & 1 \ mathbf{0} & 2 & -2 \ 0 & 4 & 1end{bmatrix}
$$
I see in my notes that elimination by multiplication yields $$
E_{21} = begin{bmatrix} 1 & 0 & 0 \ mathbf{-3} & 1 & 1 \ 0 & 0 & 1end{bmatrix}
$$.
I would like to understand the intuition behind the entries in the multiplication matrix. I am able to follow how one would find the entries for $E_{21}$ using the subtraction method (I highlighted $0$ to illustrate that I understand where that comes from, but I am supposed to use the multiplication method and I do not understand how one reaches these entries. Specifically the highlighted $-3$. Can someone please explain to me how to use the multiplication method to eliminate entries in a matrix?
matrices gaussian-elimination
matrices gaussian-elimination
edited Mar 17 at 16:35
Brian
1,208116
asked Mar 17 at 8:19
AdamAdam
345
edited Mar 17 at 16:35
Brian
1,208116
asked Mar 17 at 8:19
AdamAdam
345
edited Mar 17 at 16:35
Brian
1,208116
edited Mar 17 at 16:35
Brian
1,208116
edited Mar 17 at 16:35
Brian
1,208116
1,208116
asked Mar 17 at 8:19
AdamAdam
345
asked Mar 17 at 8:19
AdamAdam
345
asked Mar 17 at 8:19
AdamAdam
345
345
1
$begingroup$
If I understand you question correctly, you may find this article helpful. Would you mind explaining what a "multiplication matrix" is, and what the "subtraction" and "multiplications" methods to which you refer are? Is each $E_{ij}$ supposed to be an elimination matrix, or a row-equivalent matrix produced at some point during elimination? Your work seems to suggest the latter, while your notes suggest the former (both with some errors).
$endgroup$
– Brian
Mar 17 at 16:32
$begingroup$
You will also note that this very example appear in the article I linked in my previous comment.
$endgroup$
– Brian
Mar 17 at 16:35
$begingroup$
Of course, I meant to say that I am looking to find the left operand multiplied by original Matrix A that is equal to the elimination matrix E(2,1). And then, I am to eliminate entry (3,2)--which should be a "4" in my elimination matrix (2,1). I am struggling with this second step of elimination by matrix multiplication where one needs to fill in the entries for that left hand operand when setting the systems equal to the elimination matrix
$endgroup$
– Adam
Mar 17 at 17:45
add a comment |
1
$begingroup$
If I understand you question correctly, you may find this article helpful. Would you mind explaining what a "multiplication matrix" is, and what the "subtraction" and "multiplications" methods to which you refer are? Is each $E_{ij}$ supposed to be an elimination matrix, or a row-equivalent matrix produced at some point during elimination? Your work seems to suggest the latter, while your notes suggest the former (both with some errors).
$endgroup$
– Brian
Mar 17 at 16:32
$begingroup$
You will also note that this very example appear in the article I linked in my previous comment.
$endgroup$
– Brian
Mar 17 at 16:35
$begingroup$
Of course, I meant to say that I am looking to find the left operand multiplied by original Matrix A that is equal to the elimination matrix E(2,1). And then, I am to eliminate entry (3,2)--which should be a "4" in my elimination matrix (2,1). I am struggling with this second step of elimination by matrix multiplication where one needs to fill in the entries for that left hand operand when setting the systems equal to the elimination matrix
$endgroup$
– Adam
Mar 17 at 17:45
1
1
$begingroup$
If I understand you question correctly, you may find this article helpful. Would you mind explaining what a "multiplication matrix" is, and what the "subtraction" and "multiplications" methods to which you refer are? Is each $E_{ij}$ supposed to be an elimination matrix, or a row-equivalent matrix produced at some point during elimination? Your work seems to suggest the latter, while your notes suggest the former (both with some errors).
$endgroup$
– Brian
Mar 17 at 16:32
$begingroup$
If I understand you question correctly, you may find this article helpful. Would you mind explaining what a "multiplication matrix" is, and what the "subtraction" and "multiplications" methods to which you refer are? Is each $E_{ij}$ supposed to be an elimination matrix, or a row-equivalent matrix produced at some point during elimination? Your work seems to suggest the latter, while your notes suggest the former (both with some errors).
$endgroup$
– Brian
Mar 17 at 16:32
$begingroup$
You will also note that this very example appear in the article I linked in my previous comment.
$endgroup$
– Brian
Mar 17 at 16:35
$begingroup$
You will also note that this very example appear in the article I linked in my previous comment.
$endgroup$
– Brian
Mar 17 at 16:35
$begingroup$
Of course, I meant to say that I am looking to find the left operand multiplied by original Matrix A that is equal to the elimination matrix E(2,1). And then, I am to eliminate entry (3,2)--which should be a "4" in my elimination matrix (2,1). I am struggling with this second step of elimination by matrix multiplication where one needs to fill in the entries for that left hand operand when setting the systems equal to the elimination matrix
$endgroup$
– Adam
Mar 17 at 17:45
$begingroup$
Of course, I meant to say that I am looking to find the left operand multiplied by original Matrix A that is equal to the elimination matrix E(2,1). And then, I am to eliminate entry (3,2)--which should be a "4" in my elimination matrix (2,1). I am struggling with this second step of elimination by matrix multiplication where one needs to fill in the entries for that left hand operand when setting the systems equal to the elimination matrix
$endgroup$
– Adam
Mar 17 at 17:45
add a comment |
0
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$begingroup$
If I understand you question correctly, you may find this article helpful. Would you mind explaining what a "multiplication matrix" is, and what the "subtraction" and "multiplications" methods to which you refer are? Is each $E_{ij}$ supposed to be an elimination matrix, or a row-equivalent matrix produced at some point during elimination? Your work seems to suggest the latter, while your notes suggest the former (both with some errors).
$endgroup$
– Brian
Mar 17 at 16:32
$begingroup$
You will also note that this very example appear in the article I linked in my previous comment.
$endgroup$
– Brian
Mar 17 at 16:35
$begingroup$
Of course, I meant to say that I am looking to find the left operand multiplied by original Matrix A that is equal to the elimination matrix E(2,1). And then, I am to eliminate entry (3,2)--which should be a "4" in my elimination matrix (2,1). I am struggling with this second step of elimination by matrix multiplication where one needs to fill in the entries for that left hand operand when setting the systems equal to the elimination matrix
$endgroup$
– Adam
Mar 17 at 17:45