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How to denote a matrix of three indices?
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I have a "matrix" of three indices like $a_kmn$ for all $k=1,ldots,K$, $m=1,ldots,M$, and $n=1,ldots,N$ (I think it is called a tensor of order 3, right?). I denote this "matrix" by the boldface letter $mathbfA=[a_kmn]$. When I fix one "dimension" (say $k$), I write it as $mathbfA_k$ which is now the $k$th matrix of size $Mtimes N$.
My question is: are my notations fine? Or are there any standard notations used?
linear-algebra terminology definition
asked Jul 26 at 0:44
zdm
384
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I have a "matrix" of three indices like $a_kmn$ for all $k=1,ldots,K$, $m=1,ldots,M$, and $n=1,ldots,N$ (I think it is called a tensor of order 3, right?). I denote this "matrix" by the boldface letter $mathbfA=[a_kmn]$. When I fix one "dimension" (say $k$), I write it as $mathbfA_k$ which is now the $k$th matrix of size $Mtimes N$.
My question is: are my notations fine? Or are there any standard notations used?
linear-algebra terminology definition
asked Jul 26 at 0:44
zdm
384
2
Just to be aware: tensors are matrices, but they're matrices with additional properties, particularly with respect to coordinate transformations. The two terms are not completely equivalent.
– Adrian Keister
Jul 26 at 1:31
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up vote
1
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up vote
1
down vote
favorite
I have a "matrix" of three indices like $a_kmn$ for all $k=1,ldots,K$, $m=1,ldots,M$, and $n=1,ldots,N$ (I think it is called a tensor of order 3, right?). I denote this "matrix" by the boldface letter $mathbfA=[a_kmn]$. When I fix one "dimension" (say $k$), I write it as $mathbfA_k$ which is now the $k$th matrix of size $Mtimes N$.
My question is: are my notations fine? Or are there any standard notations used?
linear-algebra terminology definition
asked Jul 26 at 0:44
zdm
384
I have a "matrix" of three indices like $a_kmn$ for all $k=1,ldots,K$, $m=1,ldots,M$, and $n=1,ldots,N$ (I think it is called a tensor of order 3, right?). I denote this "matrix" by the boldface letter $mathbfA=[a_kmn]$. When I fix one "dimension" (say $k$), I write it as $mathbfA_k$ which is now the $k$th matrix of size $Mtimes N$.
My question is: are my notations fine? Or are there any standard notations used?
linear-algebra terminology definition
asked Jul 26 at 0:44
zdm
384
asked Jul 26 at 0:44
zdm
384
asked Jul 26 at 0:44
zdm
384
asked Jul 26 at 0:44
zdm
384
384
2
Just to be aware: tensors are matrices, but they're matrices with additional properties, particularly with respect to coordinate transformations. The two terms are not completely equivalent.
– Adrian Keister
Jul 26 at 1:31
add a comment |Â
2
Just to be aware: tensors are matrices, but they're matrices with additional properties, particularly with respect to coordinate transformations. The two terms are not completely equivalent.
– Adrian Keister
Jul 26 at 1:31
2
2
Just to be aware: tensors are matrices, but they're matrices with additional properties, particularly with respect to coordinate transformations. The two terms are not completely equivalent.
– Adrian Keister
Jul 26 at 1:31
Just to be aware: tensors are matrices, but they're matrices with additional properties, particularly with respect to coordinate transformations. The two terms are not completely equivalent.
– Adrian Keister
Jul 26 at 1:31
add a comment |Â
1 Answer
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oldest
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up vote
3
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Notations are... notations. Just that. If they work for you and who reads you (if it were the case), they are ok.
Going to the question of standards, I guess the name "tensor" means more than just an array (a 3D array in this case): a tensor of rank three actually can have covariant and contra-variant dimensions and that has something to do with it's physical (or mathematical) meaning and transformation rules under a change of coordinates; this has indeed consequences for the notation ($T_k^ij$, $T_jk^i$, $T^ijk$, etc.). I would save the word "tensor" to use it restrictively, depending on the context, and I would rely on the term "array" (three dimensional array, in this case) for the general case.
You can say that $A$ has components $a_ijk$ and that $Ain mathbb R^mtimes n times p$, for instance, but the notation $A_k$ for the matrix
$$Binmathbb R^mtimes n$$
with entries
$$b_ij=a_ijk,$$
I find it ambiguous, since $A_k$ or $A_i$ would always mean that you're fixing the same index. This shortcoming becames evident when you use a number, say $A_3$: whith your notation this is the matrix of componentes $a_ij3$, $1le ile m$, $1le j le n$ and there's no way to refer to a matrix of components $a_i3k$ ($1le ile m$, $1le kle p$) or the analogous version for fixed $i=3$.
Maybe you can try instead of $A_k$, something like $A_cdotcdot k$, and so: in that case $A_cdot cdot 3$ and $A_cdot 3 cdot$, for instance, would not be in general the same matrix (actually they might well be of different sizes.
Even more, you can use $A_icdot k$ or $A_ijcdot$ to get the one dimensional arrays (vectors) corresponding to fixing two indexes.
And this can actually be easily extended to $n$-dimensional arrays for arbitrary $nin mathbb N$.
NOTE: With this notation, you have $$A=A_cdotcdotcdot,$$ and of course usually you will just type $A$. Also, $$A_ijk=a_ijk$$ (or $$A_ijk=big( a_ijkbig),$$ etc.), that is, a $1times 1$ matrix, a $1$-dimensional vector or—much better— a $0$-dimensional array). That means that you need not make a difference between uppercase arrays and lowercase components (but it's not a crime either if useful).
FURTHER COMMENT on the 'inspiration' for this notation.
An situation where three or more indexes are needed arises in an experiment where measurements of a magnitude are taken for all combinations of the categories of classification of two (or more) properties (or factors, as they are called in experimental design). Supose that for every combination $(i,j)$ of the first and second factor (if we think of just two factors) there is more than one measurement and the number of repetitions is the same—say $K$—in each case: this is called a balanced design.
That is, we take the measurement $x_ijk$ which corresponds to the category $i$ for the first factor and $j$ for the second factor of classification, and it is the $k$-th repetition for that particular combination. For a balanced design of $K$ measurements for each combination of the $I$ categories in factor one and the $J$ categories in factor two, we get the $Icdot J cdot K$ measurements
$$x_ijk, quad 1le ile I,; 1le jle J,; 1le kle K,$$
each is of which is seen as a realization of the random variable $X_ijk$.
A simple example: there 6 groups of 10 people, each corresponding to the six combinations of the age factor—with categories 'adolescent', 'adult' and 'elder'— and the treatment factor ('placebo' vs. 'actual medication'), and each one's arterial pressure is measured an hour after taking the (actual or pretended) medication. In this case you have $I=3$, $J=2$ (or viceversa) and $K=10$. And $x_2,1,5$ would be the arterial pressure as measured from the fifth individual in the group of adults who are having the placebo.
In the standard model with two factors for the ANalysis Of VAriance (two-way ANOVA) the statistical model is
$$X_ijksim N(mu_ij,sigma^2),$$
all variables being independent. This can also be written
$$X_ijk=mu_ij+varepsilon_ijk,quad varepsilon_ijksim N(0,sigma^2),$$
and the $varepsilon_ijk$ variables are independent.
In general, there are further details specifying the structure of the matrix $M=big(mu_ijbig)$, such as the additive model
$$mu_ij=mu+alpha_i+beta_j, quad sum_i=1^I alpha_i=sum_j=1^J beta_j=0,$$
a model allowing for generic interactions such as
$$mu_ij=mu+alpha_i+beta_j+gamma_ij, quad sum_i=1^I alpha_i=sum_j=1^J beta_j=sum_i=1^I sum_j=1^Jgamma_ij=0,$$
which can be simplified to a multiplicative model
$$mu_ij=mu+alpha_i+beta_j+lambda alpha_i beta_j, quad sum_i=1^I alpha_i=sum_j=1^J beta_j=0,$$
etc.
So the data and the random variables $X_ijk$ form indeed a 3D-array, and when considering quantities such as sums ($S$), means ($bar X$) or number of data ($n$), the convention of the dots (or sometimes another sign instead, such as $+$ for instance), is standard notation, as in:
$$n_ijcdot:colon: textnumber of data in the 'cell' or combination $(i,j)$ of factors$$
(this is $K$ in a balanced design, but otherwise could vary among different $(i,j)$ combinations);
$$n_icdotcdot:colon: textnumber of data in the $i$-th category of the first factor of classification,$$
$$S_ijcdot=sum_k=1^K x_ijk,$$
$$S_cdot jcdot=sum_i=1^Isum_k=1^K x_ijk,$$
$$bar x_cdot jcdot=frac1n_cdot j cdotS_cdot jcdot,$$
$$bar x_ijcdot=frac1n_ijcdotS_ijcdot,$$
$$bar x_cdot cdot cdot=frac1n_cdotcdotcdotS_cdotcdotcdot,$$
and so on (the last one is sometimes called the 'great mean').
I hope all the blah helps you get a taste of how this notation works.
answered Jul 26 at 1:18
Alejandro Nasif Salum
3,07617
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
Notations are... notations. Just that. If they work for you and who reads you (if it were the case), they are ok.
Going to the question of standards, I guess the name "tensor" means more than just an array (a 3D array in this case): a tensor of rank three actually can have covariant and contra-variant dimensions and that has something to do with it's physical (or mathematical) meaning and transformation rules under a change of coordinates; this has indeed consequences for the notation ($T_k^ij$, $T_jk^i$, $T^ijk$, etc.). I would save the word "tensor" to use it restrictively, depending on the context, and I would rely on the term "array" (three dimensional array, in this case) for the general case.
You can say that $A$ has components $a_ijk$ and that $Ain mathbb R^mtimes n times p$, for instance, but the notation $A_k$ for the matrix
$$Binmathbb R^mtimes n$$
with entries
$$b_ij=a_ijk,$$
I find it ambiguous, since $A_k$ or $A_i$ would always mean that you're fixing the same index. This shortcoming becames evident when you use a number, say $A_3$: whith your notation this is the matrix of componentes $a_ij3$, $1le ile m$, $1le j le n$ and there's no way to refer to a matrix of components $a_i3k$ ($1le ile m$, $1le kle p$) or the analogous version for fixed $i=3$.
Maybe you can try instead of $A_k$, something like $A_cdotcdot k$, and so: in that case $A_cdot cdot 3$ and $A_cdot 3 cdot$, for instance, would not be in general the same matrix (actually they might well be of different sizes.
Even more, you can use $A_icdot k$ or $A_ijcdot$ to get the one dimensional arrays (vectors) corresponding to fixing two indexes.
And this can actually be easily extended to $n$-dimensional arrays for arbitrary $nin mathbb N$.
NOTE: With this notation, you have $$A=A_cdotcdotcdot,$$ and of course usually you will just type $A$. Also, $$A_ijk=a_ijk$$ (or $$A_ijk=big( a_ijkbig),$$ etc.), that is, a $1times 1$ matrix, a $1$-dimensional vector or—much better— a $0$-dimensional array). That means that you need not make a difference between uppercase arrays and lowercase components (but it's not a crime either if useful).
FURTHER COMMENT on the 'inspiration' for this notation.
An situation where three or more indexes are needed arises in an experiment where measurements of a magnitude are taken for all combinations of the categories of classification of two (or more) properties (or factors, as they are called in experimental design). Supose that for every combination $(i,j)$ of the first and second factor (if we think of just two factors) there is more than one measurement and the number of repetitions is the same—say $K$—in each case: this is called a balanced design.
That is, we take the measurement $x_ijk$ which corresponds to the category $i$ for the first factor and $j$ for the second factor of classification, and it is the $k$-th repetition for that particular combination. For a balanced design of $K$ measurements for each combination of the $I$ categories in factor one and the $J$ categories in factor two, we get the $Icdot J cdot K$ measurements
$$x_ijk, quad 1le ile I,; 1le jle J,; 1le kle K,$$
each is of which is seen as a realization of the random variable $X_ijk$.
A simple example: there 6 groups of 10 people, each corresponding to the six combinations of the age factor—with categories 'adolescent', 'adult' and 'elder'— and the treatment factor ('placebo' vs. 'actual medication'), and each one's arterial pressure is measured an hour after taking the (actual or pretended) medication. In this case you have $I=3$, $J=2$ (or viceversa) and $K=10$. And $x_2,1,5$ would be the arterial pressure as measured from the fifth individual in the group of adults who are having the placebo.
In the standard model with two factors for the ANalysis Of VAriance (two-way ANOVA) the statistical model is
$$X_ijksim N(mu_ij,sigma^2),$$
all variables being independent. This can also be written
$$X_ijk=mu_ij+varepsilon_ijk,quad varepsilon_ijksim N(0,sigma^2),$$
and the $varepsilon_ijk$ variables are independent.
In general, there are further details specifying the structure of the matrix $M=big(mu_ijbig)$, such as the additive model
$$mu_ij=mu+alpha_i+beta_j, quad sum_i=1^I alpha_i=sum_j=1^J beta_j=0,$$
a model allowing for generic interactions such as
$$mu_ij=mu+alpha_i+beta_j+gamma_ij, quad sum_i=1^I alpha_i=sum_j=1^J beta_j=sum_i=1^I sum_j=1^Jgamma_ij=0,$$
which can be simplified to a multiplicative model
$$mu_ij=mu+alpha_i+beta_j+lambda alpha_i beta_j, quad sum_i=1^I alpha_i=sum_j=1^J beta_j=0,$$
etc.
So the data and the random variables $X_ijk$ form indeed a 3D-array, and when considering quantities such as sums ($S$), means ($bar X$) or number of data ($n$), the convention of the dots (or sometimes another sign instead, such as $+$ for instance), is standard notation, as in:
$$n_ijcdot:colon: textnumber of data in the 'cell' or combination $(i,j)$ of factors$$
(this is $K$ in a balanced design, but otherwise could vary among different $(i,j)$ combinations);
$$n_icdotcdot:colon: textnumber of data in the $i$-th category of the first factor of classification,$$
$$S_ijcdot=sum_k=1^K x_ijk,$$
$$S_cdot jcdot=sum_i=1^Isum_k=1^K x_ijk,$$
$$bar x_cdot jcdot=frac1n_cdot j cdotS_cdot jcdot,$$
$$bar x_ijcdot=frac1n_ijcdotS_ijcdot,$$
$$bar x_cdot cdot cdot=frac1n_cdotcdotcdotS_cdotcdotcdot,$$
and so on (the last one is sometimes called the 'great mean').
I hope all the blah helps you get a taste of how this notation works.
answered Jul 26 at 1:18
Alejandro Nasif Salum
3,07617
add a comment |Â
up vote
3
down vote
accepted
Notations are... notations. Just that. If they work for you and who reads you (if it were the case), they are ok.
Going to the question of standards, I guess the name "tensor" means more than just an array (a 3D array in this case): a tensor of rank three actually can have covariant and contra-variant dimensions and that has something to do with it's physical (or mathematical) meaning and transformation rules under a change of coordinates; this has indeed consequences for the notation ($T_k^ij$, $T_jk^i$, $T^ijk$, etc.). I would save the word "tensor" to use it restrictively, depending on the context, and I would rely on the term "array" (three dimensional array, in this case) for the general case.
You can say that $A$ has components $a_ijk$ and that $Ain mathbb R^mtimes n times p$, for instance, but the notation $A_k$ for the matrix
$$Binmathbb R^mtimes n$$
with entries
$$b_ij=a_ijk,$$
I find it ambiguous, since $A_k$ or $A_i$ would always mean that you're fixing the same index. This shortcoming becames evident when you use a number, say $A_3$: whith your notation this is the matrix of componentes $a_ij3$, $1le ile m$, $1le j le n$ and there's no way to refer to a matrix of components $a_i3k$ ($1le ile m$, $1le kle p$) or the analogous version for fixed $i=3$.
Maybe you can try instead of $A_k$, something like $A_cdotcdot k$, and so: in that case $A_cdot cdot 3$ and $A_cdot 3 cdot$, for instance, would not be in general the same matrix (actually they might well be of different sizes.
Even more, you can use $A_icdot k$ or $A_ijcdot$ to get the one dimensional arrays (vectors) corresponding to fixing two indexes.
And this can actually be easily extended to $n$-dimensional arrays for arbitrary $nin mathbb N$.
NOTE: With this notation, you have $$A=A_cdotcdotcdot,$$ and of course usually you will just type $A$. Also, $$A_ijk=a_ijk$$ (or $$A_ijk=big( a_ijkbig),$$ etc.), that is, a $1times 1$ matrix, a $1$-dimensional vector or—much better— a $0$-dimensional array). That means that you need not make a difference between uppercase arrays and lowercase components (but it's not a crime either if useful).
FURTHER COMMENT on the 'inspiration' for this notation.
An situation where three or more indexes are needed arises in an experiment where measurements of a magnitude are taken for all combinations of the categories of classification of two (or more) properties (or factors, as they are called in experimental design). Supose that for every combination $(i,j)$ of the first and second factor (if we think of just two factors) there is more than one measurement and the number of repetitions is the same—say $K$—in each case: this is called a balanced design.
That is, we take the measurement $x_ijk$ which corresponds to the category $i$ for the first factor and $j$ for the second factor of classification, and it is the $k$-th repetition for that particular combination. For a balanced design of $K$ measurements for each combination of the $I$ categories in factor one and the $J$ categories in factor two, we get the $Icdot J cdot K$ measurements
$$x_ijk, quad 1le ile I,; 1le jle J,; 1le kle K,$$
each is of which is seen as a realization of the random variable $X_ijk$.
A simple example: there 6 groups of 10 people, each corresponding to the six combinations of the age factor—with categories 'adolescent', 'adult' and 'elder'— and the treatment factor ('placebo' vs. 'actual medication'), and each one's arterial pressure is measured an hour after taking the (actual or pretended) medication. In this case you have $I=3$, $J=2$ (or viceversa) and $K=10$. And $x_2,1,5$ would be the arterial pressure as measured from the fifth individual in the group of adults who are having the placebo.
In the standard model with two factors for the ANalysis Of VAriance (two-way ANOVA) the statistical model is
$$X_ijksim N(mu_ij,sigma^2),$$
all variables being independent. This can also be written
$$X_ijk=mu_ij+varepsilon_ijk,quad varepsilon_ijksim N(0,sigma^2),$$
and the $varepsilon_ijk$ variables are independent.
In general, there are further details specifying the structure of the matrix $M=big(mu_ijbig)$, such as the additive model
$$mu_ij=mu+alpha_i+beta_j, quad sum_i=1^I alpha_i=sum_j=1^J beta_j=0,$$
a model allowing for generic interactions such as
$$mu_ij=mu+alpha_i+beta_j+gamma_ij, quad sum_i=1^I alpha_i=sum_j=1^J beta_j=sum_i=1^I sum_j=1^Jgamma_ij=0,$$
which can be simplified to a multiplicative model
$$mu_ij=mu+alpha_i+beta_j+lambda alpha_i beta_j, quad sum_i=1^I alpha_i=sum_j=1^J beta_j=0,$$
etc.
So the data and the random variables $X_ijk$ form indeed a 3D-array, and when considering quantities such as sums ($S$), means ($bar X$) or number of data ($n$), the convention of the dots (or sometimes another sign instead, such as $+$ for instance), is standard notation, as in:
$$n_ijcdot:colon: textnumber of data in the 'cell' or combination $(i,j)$ of factors$$
(this is $K$ in a balanced design, but otherwise could vary among different $(i,j)$ combinations);
$$n_icdotcdot:colon: textnumber of data in the $i$-th category of the first factor of classification,$$
$$S_ijcdot=sum_k=1^K x_ijk,$$
$$S_cdot jcdot=sum_i=1^Isum_k=1^K x_ijk,$$
$$bar x_cdot jcdot=frac1n_cdot j cdotS_cdot jcdot,$$
$$bar x_ijcdot=frac1n_ijcdotS_ijcdot,$$
$$bar x_cdot cdot cdot=frac1n_cdotcdotcdotS_cdotcdotcdot,$$
and so on (the last one is sometimes called the 'great mean').
I hope all the blah helps you get a taste of how this notation works.
answered Jul 26 at 1:18
Alejandro Nasif Salum
3,07617
add a comment |Â
up vote
3
down vote
accepted
up vote
3
down vote
accepted
Notations are... notations. Just that. If they work for you and who reads you (if it were the case), they are ok.
Going to the question of standards, I guess the name "tensor" means more than just an array (a 3D array in this case): a tensor of rank three actually can have covariant and contra-variant dimensions and that has something to do with it's physical (or mathematical) meaning and transformation rules under a change of coordinates; this has indeed consequences for the notation ($T_k^ij$, $T_jk^i$, $T^ijk$, etc.). I would save the word "tensor" to use it restrictively, depending on the context, and I would rely on the term "array" (three dimensional array, in this case) for the general case.
You can say that $A$ has components $a_ijk$ and that $Ain mathbb R^mtimes n times p$, for instance, but the notation $A_k$ for the matrix
$$Binmathbb R^mtimes n$$
with entries
$$b_ij=a_ijk,$$
I find it ambiguous, since $A_k$ or $A_i$ would always mean that you're fixing the same index. This shortcoming becames evident when you use a number, say $A_3$: whith your notation this is the matrix of componentes $a_ij3$, $1le ile m$, $1le j le n$ and there's no way to refer to a matrix of components $a_i3k$ ($1le ile m$, $1le kle p$) or the analogous version for fixed $i=3$.
Maybe you can try instead of $A_k$, something like $A_cdotcdot k$, and so: in that case $A_cdot cdot 3$ and $A_cdot 3 cdot$, for instance, would not be in general the same matrix (actually they might well be of different sizes.
Even more, you can use $A_icdot k$ or $A_ijcdot$ to get the one dimensional arrays (vectors) corresponding to fixing two indexes.
And this can actually be easily extended to $n$-dimensional arrays for arbitrary $nin mathbb N$.
NOTE: With this notation, you have $$A=A_cdotcdotcdot,$$ and of course usually you will just type $A$. Also, $$A_ijk=a_ijk$$ (or $$A_ijk=big( a_ijkbig),$$ etc.), that is, a $1times 1$ matrix, a $1$-dimensional vector or—much better— a $0$-dimensional array). That means that you need not make a difference between uppercase arrays and lowercase components (but it's not a crime either if useful).
FURTHER COMMENT on the 'inspiration' for this notation.
An situation where three or more indexes are needed arises in an experiment where measurements of a magnitude are taken for all combinations of the categories of classification of two (or more) properties (or factors, as they are called in experimental design). Supose that for every combination $(i,j)$ of the first and second factor (if we think of just two factors) there is more than one measurement and the number of repetitions is the same—say $K$—in each case: this is called a balanced design.
That is, we take the measurement $x_ijk$ which corresponds to the category $i$ for the first factor and $j$ for the second factor of classification, and it is the $k$-th repetition for that particular combination. For a balanced design of $K$ measurements for each combination of the $I$ categories in factor one and the $J$ categories in factor two, we get the $Icdot J cdot K$ measurements
$$x_ijk, quad 1le ile I,; 1le jle J,; 1le kle K,$$
each is of which is seen as a realization of the random variable $X_ijk$.
A simple example: there 6 groups of 10 people, each corresponding to the six combinations of the age factor—with categories 'adolescent', 'adult' and 'elder'— and the treatment factor ('placebo' vs. 'actual medication'), and each one's arterial pressure is measured an hour after taking the (actual or pretended) medication. In this case you have $I=3$, $J=2$ (or viceversa) and $K=10$. And $x_2,1,5$ would be the arterial pressure as measured from the fifth individual in the group of adults who are having the placebo.
In the standard model with two factors for the ANalysis Of VAriance (two-way ANOVA) the statistical model is
$$X_ijksim N(mu_ij,sigma^2),$$
all variables being independent. This can also be written
$$X_ijk=mu_ij+varepsilon_ijk,quad varepsilon_ijksim N(0,sigma^2),$$
and the $varepsilon_ijk$ variables are independent.
In general, there are further details specifying the structure of the matrix $M=big(mu_ijbig)$, such as the additive model
$$mu_ij=mu+alpha_i+beta_j, quad sum_i=1^I alpha_i=sum_j=1^J beta_j=0,$$
a model allowing for generic interactions such as
$$mu_ij=mu+alpha_i+beta_j+gamma_ij, quad sum_i=1^I alpha_i=sum_j=1^J beta_j=sum_i=1^I sum_j=1^Jgamma_ij=0,$$
which can be simplified to a multiplicative model
$$mu_ij=mu+alpha_i+beta_j+lambda alpha_i beta_j, quad sum_i=1^I alpha_i=sum_j=1^J beta_j=0,$$
etc.
So the data and the random variables $X_ijk$ form indeed a 3D-array, and when considering quantities such as sums ($S$), means ($bar X$) or number of data ($n$), the convention of the dots (or sometimes another sign instead, such as $+$ for instance), is standard notation, as in:
$$n_ijcdot:colon: textnumber of data in the 'cell' or combination $(i,j)$ of factors$$
(this is $K$ in a balanced design, but otherwise could vary among different $(i,j)$ combinations);
$$n_icdotcdot:colon: textnumber of data in the $i$-th category of the first factor of classification,$$
$$S_ijcdot=sum_k=1^K x_ijk,$$
$$S_cdot jcdot=sum_i=1^Isum_k=1^K x_ijk,$$
$$bar x_cdot jcdot=frac1n_cdot j cdotS_cdot jcdot,$$
$$bar x_ijcdot=frac1n_ijcdotS_ijcdot,$$
$$bar x_cdot cdot cdot=frac1n_cdotcdotcdotS_cdotcdotcdot,$$
and so on (the last one is sometimes called the 'great mean').
I hope all the blah helps you get a taste of how this notation works.
answered Jul 26 at 1:18
Alejandro Nasif Salum
3,07617
Notations are... notations. Just that. If they work for you and who reads you (if it were the case), they are ok.
Going to the question of standards, I guess the name "tensor" means more than just an array (a 3D array in this case): a tensor of rank three actually can have covariant and contra-variant dimensions and that has something to do with it's physical (or mathematical) meaning and transformation rules under a change of coordinates; this has indeed consequences for the notation ($T_k^ij$, $T_jk^i$, $T^ijk$, etc.). I would save the word "tensor" to use it restrictively, depending on the context, and I would rely on the term "array" (three dimensional array, in this case) for the general case.
You can say that $A$ has components $a_ijk$ and that $Ain mathbb R^mtimes n times p$, for instance, but the notation $A_k$ for the matrix
$$Binmathbb R^mtimes n$$
with entries
$$b_ij=a_ijk,$$
I find it ambiguous, since $A_k$ or $A_i$ would always mean that you're fixing the same index. This shortcoming becames evident when you use a number, say $A_3$: whith your notation this is the matrix of componentes $a_ij3$, $1le ile m$, $1le j le n$ and there's no way to refer to a matrix of components $a_i3k$ ($1le ile m$, $1le kle p$) or the analogous version for fixed $i=3$.
Maybe you can try instead of $A_k$, something like $A_cdotcdot k$, and so: in that case $A_cdot cdot 3$ and $A_cdot 3 cdot$, for instance, would not be in general the same matrix (actually they might well be of different sizes.
Even more, you can use $A_icdot k$ or $A_ijcdot$ to get the one dimensional arrays (vectors) corresponding to fixing two indexes.
And this can actually be easily extended to $n$-dimensional arrays for arbitrary $nin mathbb N$.
NOTE: With this notation, you have $$A=A_cdotcdotcdot,$$ and of course usually you will just type $A$. Also, $$A_ijk=a_ijk$$ (or $$A_ijk=big( a_ijkbig),$$ etc.), that is, a $1times 1$ matrix, a $1$-dimensional vector or—much better— a $0$-dimensional array). That means that you need not make a difference between uppercase arrays and lowercase components (but it's not a crime either if useful).
FURTHER COMMENT on the 'inspiration' for this notation.
An situation where three or more indexes are needed arises in an experiment where measurements of a magnitude are taken for all combinations of the categories of classification of two (or more) properties (or factors, as they are called in experimental design). Supose that for every combination $(i,j)$ of the first and second factor (if we think of just two factors) there is more than one measurement and the number of repetitions is the same—say $K$—in each case: this is called a balanced design.
That is, we take the measurement $x_ijk$ which corresponds to the category $i$ for the first factor and $j$ for the second factor of classification, and it is the $k$-th repetition for that particular combination. For a balanced design of $K$ measurements for each combination of the $I$ categories in factor one and the $J$ categories in factor two, we get the $Icdot J cdot K$ measurements
$$x_ijk, quad 1le ile I,; 1le jle J,; 1le kle K,$$
each is of which is seen as a realization of the random variable $X_ijk$.
A simple example: there 6 groups of 10 people, each corresponding to the six combinations of the age factor—with categories 'adolescent', 'adult' and 'elder'— and the treatment factor ('placebo' vs. 'actual medication'), and each one's arterial pressure is measured an hour after taking the (actual or pretended) medication. In this case you have $I=3$, $J=2$ (or viceversa) and $K=10$. And $x_2,1,5$ would be the arterial pressure as measured from the fifth individual in the group of adults who are having the placebo.
In the standard model with two factors for the ANalysis Of VAriance (two-way ANOVA) the statistical model is
$$X_ijksim N(mu_ij,sigma^2),$$
all variables being independent. This can also be written
$$X_ijk=mu_ij+varepsilon_ijk,quad varepsilon_ijksim N(0,sigma^2),$$
and the $varepsilon_ijk$ variables are independent.
In general, there are further details specifying the structure of the matrix $M=big(mu_ijbig)$, such as the additive model
$$mu_ij=mu+alpha_i+beta_j, quad sum_i=1^I alpha_i=sum_j=1^J beta_j=0,$$
a model allowing for generic interactions such as
$$mu_ij=mu+alpha_i+beta_j+gamma_ij, quad sum_i=1^I alpha_i=sum_j=1^J beta_j=sum_i=1^I sum_j=1^Jgamma_ij=0,$$
which can be simplified to a multiplicative model
$$mu_ij=mu+alpha_i+beta_j+lambda alpha_i beta_j, quad sum_i=1^I alpha_i=sum_j=1^J beta_j=0,$$
etc.
So the data and the random variables $X_ijk$ form indeed a 3D-array, and when considering quantities such as sums ($S$), means ($bar X$) or number of data ($n$), the convention of the dots (or sometimes another sign instead, such as $+$ for instance), is standard notation, as in:
$$n_ijcdot:colon: textnumber of data in the 'cell' or combination $(i,j)$ of factors$$
(this is $K$ in a balanced design, but otherwise could vary among different $(i,j)$ combinations);
$$n_icdotcdot:colon: textnumber of data in the $i$-th category of the first factor of classification,$$
$$S_ijcdot=sum_k=1^K x_ijk,$$
$$S_cdot jcdot=sum_i=1^Isum_k=1^K x_ijk,$$
$$bar x_cdot jcdot=frac1n_cdot j cdotS_cdot jcdot,$$
$$bar x_ijcdot=frac1n_ijcdotS_ijcdot,$$
$$bar x_cdot cdot cdot=frac1n_cdotcdotcdotS_cdotcdotcdot,$$
and so on (the last one is sometimes called the 'great mean').
I hope all the blah helps you get a taste of how this notation works.
answered Jul 26 at 1:18
Alejandro Nasif Salum
3,07617
edited Jul 26 at 18:52
answered Jul 26 at 1:18
Alejandro Nasif Salum
3,07617
answered Jul 26 at 1:18
Alejandro Nasif Salum
3,07617
answered Jul 26 at 1:18
Alejandro Nasif Salum
3,07617
3,07617
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2
Just to be aware: tensors are matrices, but they're matrices with additional properties, particularly with respect to coordinate transformations. The two terms are not completely equivalent.
– Adrian Keister
Jul 26 at 1:31