Terminology: Boolean Matrix that depends on a Boolean Sequence for its values?
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I've been studying certain types of statements and in doing so found an interesting (to me anyway) structure. I'm not a mathematician by trade, so I find it difficult to find my way around the literature to find answers to my questions. I'm hoping someone here may have found this object during their studies.
Let $S$ be a binary sequence and $s_i,s_j$ be the $i$-th and $j$-th value, respectively, of the sequence $S$.
Construct a matrix with the form:
$$
begin{bmatrix}
s_0land s_0 & s_0land s_1 & s_0land s_2 & cdots & s_0land s_j \
0 & s_1land s_0 & s_1land s_1 & cdots & s_1land s_{j-1} \
0 & 0 & s_2land s_0 & cdots & s_2land s_{j-2} \
vdots & vdots & vdots & ddots & vdots \
0 & 0 & 0 & 0 & s_iland s_0 \
end{bmatrix}
$$
Example:
$$
S=111011001\
begin{bmatrix}
1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 \
0 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 \
0 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 1 \
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1
end{bmatrix}
$$
I looked through the Wikipedia article that lists the different kinds of matrices and didn't find any, aside from the most basic ones like "(0,1) matrices", that don't quite fit the structure that I have here (unless I misunderstood some of them). I've also looked at the "The Matrix Cookbook" (v2012.11.15) and couldn't find what I was looking for there either.
To me it represents the solution space of certain statements up to some value $N$.
The example posted above represents the solution space (with repeated solutions "(0, 1)" and "(1, 0)" for example) for $0 leq n leq 8$ to the statement "Every non-negative integer is the sum of 4 square numbers." where the sequence $S$ represents whether a number is the sum of 2 square numbers. The columns of the matrix, whenever there is a 1, represents a solution.
My questions are:
- What is this kind of matrix usually called?
- What branches of mathematics should I be looking at to learn more about it?
linear-algebra terminology boolean-algebra binary
asked Apr 30 '18 at 17:04
Jonatas MiguelJonatas Miguel
65
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add a comment |
$begingroup$
I've been studying certain types of statements and in doing so found an interesting (to me anyway) structure. I'm not a mathematician by trade, so I find it difficult to find my way around the literature to find answers to my questions. I'm hoping someone here may have found this object during their studies.
Let $S$ be a binary sequence and $s_i,s_j$ be the $i$-th and $j$-th value, respectively, of the sequence $S$.
Construct a matrix with the form:
$$
begin{bmatrix}
s_0land s_0 & s_0land s_1 & s_0land s_2 & cdots & s_0land s_j \
0 & s_1land s_0 & s_1land s_1 & cdots & s_1land s_{j-1} \
0 & 0 & s_2land s_0 & cdots & s_2land s_{j-2} \
vdots & vdots & vdots & ddots & vdots \
0 & 0 & 0 & 0 & s_iland s_0 \
end{bmatrix}
$$
Example:
$$
S=111011001\
begin{bmatrix}
1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 \
0 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 \
0 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 1 \
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1
end{bmatrix}
$$
I looked through the Wikipedia article that lists the different kinds of matrices and didn't find any, aside from the most basic ones like "(0,1) matrices", that don't quite fit the structure that I have here (unless I misunderstood some of them). I've also looked at the "The Matrix Cookbook" (v2012.11.15) and couldn't find what I was looking for there either.
To me it represents the solution space of certain statements up to some value $N$.
The example posted above represents the solution space (with repeated solutions "(0, 1)" and "(1, 0)" for example) for $0 leq n leq 8$ to the statement "Every non-negative integer is the sum of 4 square numbers." where the sequence $S$ represents whether a number is the sum of 2 square numbers. The columns of the matrix, whenever there is a 1, represents a solution.
My questions are:
- What is this kind of matrix usually called?
- What branches of mathematics should I be looking at to learn more about it?
linear-algebra terminology boolean-algebra binary
asked Apr 30 '18 at 17:04
Jonatas MiguelJonatas Miguel
65
$endgroup$
add a comment |
$begingroup$
I've been studying certain types of statements and in doing so found an interesting (to me anyway) structure. I'm not a mathematician by trade, so I find it difficult to find my way around the literature to find answers to my questions. I'm hoping someone here may have found this object during their studies.
Let $S$ be a binary sequence and $s_i,s_j$ be the $i$-th and $j$-th value, respectively, of the sequence $S$.
Construct a matrix with the form:
$$
begin{bmatrix}
s_0land s_0 & s_0land s_1 & s_0land s_2 & cdots & s_0land s_j \
0 & s_1land s_0 & s_1land s_1 & cdots & s_1land s_{j-1} \
0 & 0 & s_2land s_0 & cdots & s_2land s_{j-2} \
vdots & vdots & vdots & ddots & vdots \
0 & 0 & 0 & 0 & s_iland s_0 \
end{bmatrix}
$$
Example:
$$
S=111011001\
begin{bmatrix}
1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 \
0 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 \
0 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 1 \
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1
end{bmatrix}
$$
I looked through the Wikipedia article that lists the different kinds of matrices and didn't find any, aside from the most basic ones like "(0,1) matrices", that don't quite fit the structure that I have here (unless I misunderstood some of them). I've also looked at the "The Matrix Cookbook" (v2012.11.15) and couldn't find what I was looking for there either.
To me it represents the solution space of certain statements up to some value $N$.
The example posted above represents the solution space (with repeated solutions "(0, 1)" and "(1, 0)" for example) for $0 leq n leq 8$ to the statement "Every non-negative integer is the sum of 4 square numbers." where the sequence $S$ represents whether a number is the sum of 2 square numbers. The columns of the matrix, whenever there is a 1, represents a solution.
My questions are:
- What is this kind of matrix usually called?
- What branches of mathematics should I be looking at to learn more about it?
linear-algebra terminology boolean-algebra binary
asked Apr 30 '18 at 17:04
Jonatas MiguelJonatas Miguel
65
$endgroup$
I've been studying certain types of statements and in doing so found an interesting (to me anyway) structure. I'm not a mathematician by trade, so I find it difficult to find my way around the literature to find answers to my questions. I'm hoping someone here may have found this object during their studies.
Let $S$ be a binary sequence and $s_i,s_j$ be the $i$-th and $j$-th value, respectively, of the sequence $S$.
Construct a matrix with the form:
$$
begin{bmatrix}
s_0land s_0 & s_0land s_1 & s_0land s_2 & cdots & s_0land s_j \
0 & s_1land s_0 & s_1land s_1 & cdots & s_1land s_{j-1} \
0 & 0 & s_2land s_0 & cdots & s_2land s_{j-2} \
vdots & vdots & vdots & ddots & vdots \
0 & 0 & 0 & 0 & s_iland s_0 \
end{bmatrix}
$$
Example:
$$
S=111011001\
begin{bmatrix}
1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 \
0 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 \
0 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 1 \
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1
end{bmatrix}
$$
I looked through the Wikipedia article that lists the different kinds of matrices and didn't find any, aside from the most basic ones like "(0,1) matrices", that don't quite fit the structure that I have here (unless I misunderstood some of them). I've also looked at the "The Matrix Cookbook" (v2012.11.15) and couldn't find what I was looking for there either.
To me it represents the solution space of certain statements up to some value $N$.
The example posted above represents the solution space (with repeated solutions "(0, 1)" and "(1, 0)" for example) for $0 leq n leq 8$ to the statement "Every non-negative integer is the sum of 4 square numbers." where the sequence $S$ represents whether a number is the sum of 2 square numbers. The columns of the matrix, whenever there is a 1, represents a solution.
My questions are:
- What is this kind of matrix usually called?
- What branches of mathematics should I be looking at to learn more about it?
linear-algebra terminology boolean-algebra binary
linear-algebra terminology boolean-algebra binary
asked Apr 30 '18 at 17:04
Jonatas MiguelJonatas Miguel
65
asked Apr 30 '18 at 17:04
Jonatas MiguelJonatas Miguel
65
edited Dec 17 '18 at 19:42
Jonatas Miguel
asked Apr 30 '18 at 17:04
Jonatas MiguelJonatas Miguel
65
asked Apr 30 '18 at 17:04
Jonatas MiguelJonatas Miguel
65
asked Apr 30 '18 at 17:04
Jonatas MiguelJonatas Miguel
65
65
add a comment |
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