How do I complete a matrix so that its columns are orthogonal? The 2019 Stack Overflow Developer Survey Results Are InIs the matrix in the given dot product orthogonalMatrix columns and independenceProof that columns of an invertible matrix are linearly independentIf $P$ is an orthogonal matrix its columns form an orthonormal set.Generate examples for the intersection of 3 planesHow to change the third column of this matrix so that it becomes orthogonal?Dot product of a matrix columns with vector to show no solutions to $Acdot x = b$Simplify the matrix of a linear system knowing that some of the solutions are equalConvert matrix columns to z scores - notationLinear Algebra, orthogonal columns and length
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How do I complete a matrix so that its columns are orthogonal?
The 2019 Stack Overflow Developer Survey Results Are InIs the matrix in the given dot product orthogonalMatrix columns and independenceProof that columns of an invertible matrix are linearly independentIf $P$ is an orthogonal matrix its columns form an orthonormal set.Generate examples for the intersection of 3 planesHow to change the third column of this matrix so that it becomes orthogonal?Dot product of a matrix columns with vector to show no solutions to $Acdot x = b$Simplify the matrix of a linear system knowing that some of the solutions are equalConvert matrix columns to z scores - notationLinear Algebra, orthogonal columns and length
$begingroup$
I'm not sure how to start on these types of problems. I know that the dot product of the columns have to be zero between each pair.
Do I set up a system of linear equations for this? If so, what would it even look like?
linear-algebra matrices orthogonal-matrices matrix-completion
edited Mar 30 at 18:28
Rodrigo de Azevedo
13.2k41962
asked Mar 30 at 17:32
Samurai BaleSamurai Bale
726
$endgroup$
add a comment |
$begingroup$
I'm not sure how to start on these types of problems. I know that the dot product of the columns have to be zero between each pair.
Do I set up a system of linear equations for this? If so, what would it even look like?
linear-algebra matrices orthogonal-matrices matrix-completion
edited Mar 30 at 18:28
Rodrigo de Azevedo
13.2k41962
asked Mar 30 at 17:32
Samurai BaleSamurai Bale
726
$endgroup$
2
$begingroup$
This is a great problem. It gives me ideas for my own course.
$endgroup$
– James S. Cook
Mar 30 at 17:39
1
$begingroup$
Remember that two columns are orthogonal if and only if their dot product is zero. For matrix A what you need to do is find a, b, c, d so that the dot product is zero pairwise for all columns. And something similar for matrix B.
$endgroup$
– ErotemeObelus
Mar 30 at 18:19
add a comment |
$begingroup$
I'm not sure how to start on these types of problems. I know that the dot product of the columns have to be zero between each pair.
Do I set up a system of linear equations for this? If so, what would it even look like?
linear-algebra matrices orthogonal-matrices matrix-completion
edited Mar 30 at 18:28
Rodrigo de Azevedo
13.2k41962
asked Mar 30 at 17:32
Samurai BaleSamurai Bale
726
$endgroup$
I'm not sure how to start on these types of problems. I know that the dot product of the columns have to be zero between each pair.
Do I set up a system of linear equations for this? If so, what would it even look like?
linear-algebra matrices orthogonal-matrices matrix-completion
linear-algebra matrices orthogonal-matrices matrix-completion
edited Mar 30 at 18:28
Rodrigo de Azevedo
13.2k41962
asked Mar 30 at 17:32
Samurai BaleSamurai Bale
726
edited Mar 30 at 18:28
Rodrigo de Azevedo
13.2k41962
asked Mar 30 at 17:32
Samurai BaleSamurai Bale
726
edited Mar 30 at 18:28
Rodrigo de Azevedo
13.2k41962
edited Mar 30 at 18:28
Rodrigo de Azevedo
13.2k41962
edited Mar 30 at 18:28
Rodrigo de Azevedo
13.2k41962
13.2k41962
asked Mar 30 at 17:32
Samurai BaleSamurai Bale
726
asked Mar 30 at 17:32
Samurai BaleSamurai Bale
726
asked Mar 30 at 17:32
Samurai BaleSamurai Bale
726
726
2
$begingroup$
This is a great problem. It gives me ideas for my own course.
$endgroup$
– James S. Cook
Mar 30 at 17:39
1
$begingroup$
Remember that two columns are orthogonal if and only if their dot product is zero. For matrix A what you need to do is find a, b, c, d so that the dot product is zero pairwise for all columns. And something similar for matrix B.
$endgroup$
– ErotemeObelus
Mar 30 at 18:19
add a comment |
2
$begingroup$
This is a great problem. It gives me ideas for my own course.
$endgroup$
– James S. Cook
Mar 30 at 17:39
1
$begingroup$
Remember that two columns are orthogonal if and only if their dot product is zero. For matrix A what you need to do is find a, b, c, d so that the dot product is zero pairwise for all columns. And something similar for matrix B.
$endgroup$
– ErotemeObelus
Mar 30 at 18:19
2
2
$begingroup$
This is a great problem. It gives me ideas for my own course.
$endgroup$
– James S. Cook
Mar 30 at 17:39
$begingroup$
This is a great problem. It gives me ideas for my own course.
$endgroup$
– James S. Cook
Mar 30 at 17:39
1
1
$begingroup$
Remember that two columns are orthogonal if and only if their dot product is zero. For matrix A what you need to do is find a, b, c, d so that the dot product is zero pairwise for all columns. And something similar for matrix B.
$endgroup$
– ErotemeObelus
Mar 30 at 18:19
$begingroup$
Remember that two columns are orthogonal if and only if their dot product is zero. For matrix A what you need to do is find a, b, c, d so that the dot product is zero pairwise for all columns. And something similar for matrix B.
$endgroup$
– ErotemeObelus
Mar 30 at 18:19
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Your matrix $A$ is$$A=beginbmatrix-1&1&-1&1\1&1&-1&a\0&1&b&c\0&0&1&dendbmatrix.$$The first to columns are clearly orthogonal. Also, the first and the third columns are orthogonal too, no matter what $B$ is. On the other hand, the second and the second columns are orthogonal if and only if $1times(-1)+1times(-1)+1times b+0times1=0$, which means that $b=2$. Let us then assume that $b=2$
Now, the first and the fourth columns are orthogonal if and only if $a=1$. So let us assume that $a=1$. Then the second and the fourth columns are orthogonal if and only if $c=-2$. Now, complete the process in order to determine the only possible value for $d$. Can you deal with the other matrix now?
answered Mar 30 at 17:40
José Carlos SantosJosé Carlos Santos
174k23133242
$endgroup$
$begingroup$
for the second matrix the first column's missing number is -1 and the third column is 1 ,-1 -,1, is there a step by step method to solve these types of problems
$endgroup$
– Samurai Bale
Mar 30 at 18:02
$begingroup$
The method that I described is step-by-step.
$endgroup$
– José Carlos Santos
Mar 30 at 18:04
add a comment |
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$begingroup$
Your matrix $A$ is$$A=beginbmatrix-1&1&-1&1\1&1&-1&a\0&1&b&c\0&0&1&dendbmatrix.$$The first to columns are clearly orthogonal. Also, the first and the third columns are orthogonal too, no matter what $B$ is. On the other hand, the second and the second columns are orthogonal if and only if $1times(-1)+1times(-1)+1times b+0times1=0$, which means that $b=2$. Let us then assume that $b=2$
Now, the first and the fourth columns are orthogonal if and only if $a=1$. So let us assume that $a=1$. Then the second and the fourth columns are orthogonal if and only if $c=-2$. Now, complete the process in order to determine the only possible value for $d$. Can you deal with the other matrix now?
answered Mar 30 at 17:40
José Carlos SantosJosé Carlos Santos
174k23133242
$endgroup$
$begingroup$
for the second matrix the first column's missing number is -1 and the third column is 1 ,-1 -,1, is there a step by step method to solve these types of problems
$endgroup$
– Samurai Bale
Mar 30 at 18:02
$begingroup$
The method that I described is step-by-step.
$endgroup$
– José Carlos Santos
Mar 30 at 18:04
add a comment |
$begingroup$
Your matrix $A$ is$$A=beginbmatrix-1&1&-1&1\1&1&-1&a\0&1&b&c\0&0&1&dendbmatrix.$$The first to columns are clearly orthogonal. Also, the first and the third columns are orthogonal too, no matter what $B$ is. On the other hand, the second and the second columns are orthogonal if and only if $1times(-1)+1times(-1)+1times b+0times1=0$, which means that $b=2$. Let us then assume that $b=2$
Now, the first and the fourth columns are orthogonal if and only if $a=1$. So let us assume that $a=1$. Then the second and the fourth columns are orthogonal if and only if $c=-2$. Now, complete the process in order to determine the only possible value for $d$. Can you deal with the other matrix now?
answered Mar 30 at 17:40
José Carlos SantosJosé Carlos Santos
174k23133242
$endgroup$
$begingroup$
for the second matrix the first column's missing number is -1 and the third column is 1 ,-1 -,1, is there a step by step method to solve these types of problems
$endgroup$
– Samurai Bale
Mar 30 at 18:02
$begingroup$
The method that I described is step-by-step.
$endgroup$
– José Carlos Santos
Mar 30 at 18:04
add a comment |
$begingroup$
Your matrix $A$ is$$A=beginbmatrix-1&1&-1&1\1&1&-1&a\0&1&b&c\0&0&1&dendbmatrix.$$The first to columns are clearly orthogonal. Also, the first and the third columns are orthogonal too, no matter what $B$ is. On the other hand, the second and the second columns are orthogonal if and only if $1times(-1)+1times(-1)+1times b+0times1=0$, which means that $b=2$. Let us then assume that $b=2$
Now, the first and the fourth columns are orthogonal if and only if $a=1$. So let us assume that $a=1$. Then the second and the fourth columns are orthogonal if and only if $c=-2$. Now, complete the process in order to determine the only possible value for $d$. Can you deal with the other matrix now?
answered Mar 30 at 17:40
José Carlos SantosJosé Carlos Santos
174k23133242
$endgroup$
Your matrix $A$ is$$A=beginbmatrix-1&1&-1&1\1&1&-1&a\0&1&b&c\0&0&1&dendbmatrix.$$The first to columns are clearly orthogonal. Also, the first and the third columns are orthogonal too, no matter what $B$ is. On the other hand, the second and the second columns are orthogonal if and only if $1times(-1)+1times(-1)+1times b+0times1=0$, which means that $b=2$. Let us then assume that $b=2$
Now, the first and the fourth columns are orthogonal if and only if $a=1$. So let us assume that $a=1$. Then the second and the fourth columns are orthogonal if and only if $c=-2$. Now, complete the process in order to determine the only possible value for $d$. Can you deal with the other matrix now?
answered Mar 30 at 17:40
José Carlos SantosJosé Carlos Santos
174k23133242
answered Mar 30 at 17:40
José Carlos SantosJosé Carlos Santos
174k23133242
answered Mar 30 at 17:40
José Carlos SantosJosé Carlos Santos
174k23133242
answered Mar 30 at 17:40
José Carlos SantosJosé Carlos Santos
174k23133242
174k23133242
$begingroup$
for the second matrix the first column's missing number is -1 and the third column is 1 ,-1 -,1, is there a step by step method to solve these types of problems
$endgroup$
– Samurai Bale
Mar 30 at 18:02
$begingroup$
The method that I described is step-by-step.
$endgroup$
– José Carlos Santos
Mar 30 at 18:04
add a comment |
$begingroup$
for the second matrix the first column's missing number is -1 and the third column is 1 ,-1 -,1, is there a step by step method to solve these types of problems
$endgroup$
– Samurai Bale
Mar 30 at 18:02
$begingroup$
The method that I described is step-by-step.
$endgroup$
– José Carlos Santos
Mar 30 at 18:04
$begingroup$
for the second matrix the first column's missing number is -1 and the third column is 1 ,-1 -,1, is there a step by step method to solve these types of problems
$endgroup$
– Samurai Bale
Mar 30 at 18:02
$begingroup$
for the second matrix the first column's missing number is -1 and the third column is 1 ,-1 -,1, is there a step by step method to solve these types of problems
$endgroup$
– Samurai Bale
Mar 30 at 18:02
$begingroup$
The method that I described is step-by-step.
$endgroup$
– José Carlos Santos
Mar 30 at 18:04
$begingroup$
The method that I described is step-by-step.
$endgroup$
– José Carlos Santos
Mar 30 at 18:04
add a comment |
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2
$begingroup$
This is a great problem. It gives me ideas for my own course.
$endgroup$
– James S. Cook
Mar 30 at 17:39
1
$begingroup$
Remember that two columns are orthogonal if and only if their dot product is zero. For matrix A what you need to do is find a, b, c, d so that the dot product is zero pairwise for all columns. And something similar for matrix B.
$endgroup$
– ErotemeObelus
Mar 30 at 18:19