Mixing
Is there a special name for the orthogonal projection matrix onto the unit vector?
$begingroup$
Many problems in multivariate analysis involve the $n times n$ matrix:
$$mathbf{M} equiv boldsymbol{I}_n - frac{1}{n} mathbf{1}_{n times n}.$$
This is an orthogonal projection matrix onto the unit vector, so when it is applied to a column vector containing values $Y_1,...,Y_n$, this matrix subtracts the sample mean $bar{Y}_n$ from these values:
$$mathbf{Y} = begin{bmatrix} Y_1 \ vdots \ Y_n end{bmatrix}
quad quad quad implies quad quad quad
mathbf{M} mathbf{Y} = begin{bmatrix} Y_1 - bar{Y}_n \ vdots \ Y_n - bar{Y}_n end{bmatrix}.$$
There are many simple properties of this matrix, owing to the fact that it is an orthogonal projection matrix. It has $text{tr}(mathbf{M}) = text{rank}(mathbf{M}) = n-1$, which means that it has a single zero eigenvalue and the remaining eigenvalues are all ones. It comes up a lot in multivariate analysis, including statistical problems, where it is common to look at random vectors after subtracting their sample means.
My question: Does this matrix have any special name? Is there a literature on this type of matrix? Aside from the properties I have listed here, are there any other important properties of this matrix that are useful in statistical problems dealing with random vectors?
matrices terminology projection
asked Jan 31 at 2:20
BenBen
1,900215
$endgroup$
add a comment |
$begingroup$
Many problems in multivariate analysis involve the $n times n$ matrix:
$$mathbf{M} equiv boldsymbol{I}_n - frac{1}{n} mathbf{1}_{n times n}.$$
This is an orthogonal projection matrix onto the unit vector, so when it is applied to a column vector containing values $Y_1,...,Y_n$, this matrix subtracts the sample mean $bar{Y}_n$ from these values:
$$mathbf{Y} = begin{bmatrix} Y_1 \ vdots \ Y_n end{bmatrix}
quad quad quad implies quad quad quad
mathbf{M} mathbf{Y} = begin{bmatrix} Y_1 - bar{Y}_n \ vdots \ Y_n - bar{Y}_n end{bmatrix}.$$
There are many simple properties of this matrix, owing to the fact that it is an orthogonal projection matrix. It has $text{tr}(mathbf{M}) = text{rank}(mathbf{M}) = n-1$, which means that it has a single zero eigenvalue and the remaining eigenvalues are all ones. It comes up a lot in multivariate analysis, including statistical problems, where it is common to look at random vectors after subtracting their sample means.
My question: Does this matrix have any special name? Is there a literature on this type of matrix? Aside from the properties I have listed here, are there any other important properties of this matrix that are useful in statistical problems dealing with random vectors?
matrices terminology projection
asked Jan 31 at 2:20
BenBen
1,900215
$endgroup$
$begingroup$
Maybe look at MDS or PCA literature. I vaguely recall it being called the mean-centering matrix there.
$endgroup$
– obscurans
Jan 31 at 2:59
add a comment |
$begingroup$
Many problems in multivariate analysis involve the $n times n$ matrix:
$$mathbf{M} equiv boldsymbol{I}_n - frac{1}{n} mathbf{1}_{n times n}.$$
This is an orthogonal projection matrix onto the unit vector, so when it is applied to a column vector containing values $Y_1,...,Y_n$, this matrix subtracts the sample mean $bar{Y}_n$ from these values:
$$mathbf{Y} = begin{bmatrix} Y_1 \ vdots \ Y_n end{bmatrix}
quad quad quad implies quad quad quad
mathbf{M} mathbf{Y} = begin{bmatrix} Y_1 - bar{Y}_n \ vdots \ Y_n - bar{Y}_n end{bmatrix}.$$
There are many simple properties of this matrix, owing to the fact that it is an orthogonal projection matrix. It has $text{tr}(mathbf{M}) = text{rank}(mathbf{M}) = n-1$, which means that it has a single zero eigenvalue and the remaining eigenvalues are all ones. It comes up a lot in multivariate analysis, including statistical problems, where it is common to look at random vectors after subtracting their sample means.
My question: Does this matrix have any special name? Is there a literature on this type of matrix? Aside from the properties I have listed here, are there any other important properties of this matrix that are useful in statistical problems dealing with random vectors?
matrices terminology projection
asked Jan 31 at 2:20
BenBen
1,900215
$endgroup$
Many problems in multivariate analysis involve the $n times n$ matrix:
$$mathbf{M} equiv boldsymbol{I}_n - frac{1}{n} mathbf{1}_{n times n}.$$
This is an orthogonal projection matrix onto the unit vector, so when it is applied to a column vector containing values $Y_1,...,Y_n$, this matrix subtracts the sample mean $bar{Y}_n$ from these values:
$$mathbf{Y} = begin{bmatrix} Y_1 \ vdots \ Y_n end{bmatrix}
quad quad quad implies quad quad quad
mathbf{M} mathbf{Y} = begin{bmatrix} Y_1 - bar{Y}_n \ vdots \ Y_n - bar{Y}_n end{bmatrix}.$$
There are many simple properties of this matrix, owing to the fact that it is an orthogonal projection matrix. It has $text{tr}(mathbf{M}) = text{rank}(mathbf{M}) = n-1$, which means that it has a single zero eigenvalue and the remaining eigenvalues are all ones. It comes up a lot in multivariate analysis, including statistical problems, where it is common to look at random vectors after subtracting their sample means.
My question: Does this matrix have any special name? Is there a literature on this type of matrix? Aside from the properties I have listed here, are there any other important properties of this matrix that are useful in statistical problems dealing with random vectors?
matrices terminology projection
matrices terminology projection
asked Jan 31 at 2:20
BenBen
1,900215
asked Jan 31 at 2:20
BenBen
1,900215
asked Jan 31 at 2:20
BenBen
1,900215
asked Jan 31 at 2:20
BenBen
1,900215
asked Jan 31 at 2:20
BenBen
1,900215
1,900215
$begingroup$
Maybe look at MDS or PCA literature. I vaguely recall it being called the mean-centering matrix there.
$endgroup$
– obscurans
Jan 31 at 2:59
add a comment |
$begingroup$
Maybe look at MDS or PCA literature. I vaguely recall it being called the mean-centering matrix there.
$endgroup$
– obscurans
Jan 31 at 2:59
$begingroup$
Maybe look at MDS or PCA literature. I vaguely recall it being called the mean-centering matrix there.
$endgroup$
– obscurans
Jan 31 at 2:59
$begingroup$
Maybe look at MDS or PCA literature. I vaguely recall it being called the mean-centering matrix there.
$endgroup$
– obscurans
Jan 31 at 2:59
add a comment |
1 Answer
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$begingroup$
From the comment by obscurans I have now found out that this is called the centering matrix. It appears to be discussed specifically in some matrix textbooks, such as in Zhang (2017) (pp. 111-112). It does not appear to have any other important properties beyond what was listed in the question.
answered Jan 31 at 3:08
BenBen
1,900215
$endgroup$
add a comment |
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1 Answer
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$begingroup$
From the comment by obscurans I have now found out that this is called the centering matrix. It appears to be discussed specifically in some matrix textbooks, such as in Zhang (2017) (pp. 111-112). It does not appear to have any other important properties beyond what was listed in the question.
answered Jan 31 at 3:08
BenBen
1,900215
$endgroup$
add a comment |
$begingroup$
From the comment by obscurans I have now found out that this is called the centering matrix. It appears to be discussed specifically in some matrix textbooks, such as in Zhang (2017) (pp. 111-112). It does not appear to have any other important properties beyond what was listed in the question.
answered Jan 31 at 3:08
BenBen
1,900215
$endgroup$
add a comment |
$begingroup$
From the comment by obscurans I have now found out that this is called the centering matrix. It appears to be discussed specifically in some matrix textbooks, such as in Zhang (2017) (pp. 111-112). It does not appear to have any other important properties beyond what was listed in the question.
answered Jan 31 at 3:08
BenBen
1,900215
$endgroup$
From the comment by obscurans I have now found out that this is called the centering matrix. It appears to be discussed specifically in some matrix textbooks, such as in Zhang (2017) (pp. 111-112). It does not appear to have any other important properties beyond what was listed in the question.
answered Jan 31 at 3:08
BenBen
1,900215
answered Jan 31 at 3:08
BenBen
1,900215
answered Jan 31 at 3:08
BenBen
1,900215
answered Jan 31 at 3:08
BenBen
1,900215
1,900215
add a comment |
add a comment |
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$begingroup$
Maybe look at MDS or PCA literature. I vaguely recall it being called the mean-centering matrix there.
$endgroup$
– obscurans
Jan 31 at 2:59