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Changed tags and title to reflect that the question is about a mistake by Pearson, rather than some terminology difference (as I originally though)
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Kimmel
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Terminology: matrix minors and cofactors (involves Cramer's rule) Mistake in Karl Pearson's 1900 paper introducing the chi-squared distribution

removed capitals from title
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YCor
YCor
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Terminology: Matrix Minorsmatrix minors and Cofactorscofactors (involves Cramer's Rulerule)

Background

I'm reading Karl Pearson's 1900 paper titled:

On the Criterioncriterion that a given Systemsystem of Deviationsdeviations from the Probableprobable in the Casecase of a Correlated Systemcorrelated system of Variablesvariables is such that it can be reasonably supposed to have arisen from Random Samplingrandom sampling

I paraphrase the opening. He supposes $x_1, x_2, \ldots, x_n$ is a system of random variables following a multivariate normal distribution centered at the origin. Then he asserts the probability density of the vector $\langle x_1, x_2, \ldots, x_n \rangle$ is $$ Z = Z_0 \exp\left( -\frac{1}{2\det S}\sum_i \sum_j S_{pq} x_p x_q\right)\,, $$ where

  • $S$ is the covariance matrix of $x_1, x_2, \ldots, x_n$
  • $S_{pq}$ is the "minor obtained by striking out the $p$th row and $q$th column" of the covariance matrix
  • $Z_0$ is an unspecified constant.

This appears inconsistent with the multivariate normal density function from Wikipedia, which involves a matrix inverse. These two expressions for the density function can be reconciled using Cramer's rule, but only if what Karl Pearson meant by "minor" is what we today would call a "cofactor."

Question

In 1900, did the word "minor" refer to what we would call a "cofactor"?

Terminology: Matrix Minors and Cofactors (involves Cramer's Rule)

Background

I'm reading Karl Pearson's 1900 paper titled:

On the Criterion that a given System of Deviations from the Probable in the Case of a Correlated System of Variables is such that it can be reasonably supposed to have arisen from Random Sampling

I paraphrase the opening. He supposes $x_1, x_2, \ldots, x_n$ is a system of random variables following a multivariate normal distribution centered at the origin. Then he asserts the probability density of the vector $\langle x_1, x_2, \ldots, x_n \rangle$ is $$ Z = Z_0 \exp\left( -\frac{1}{2\det S}\sum_i \sum_j S_{pq} x_p x_q\right)\,, $$ where

  • $S$ is the covariance matrix of $x_1, x_2, \ldots, x_n$
  • $S_{pq}$ is the "minor obtained by striking out the $p$th row and $q$th column" of the covariance matrix
  • $Z_0$ is an unspecified constant.

This appears inconsistent with the multivariate normal density function from Wikipedia, which involves a matrix inverse. These two expressions for the density function can be reconciled using Cramer's rule, but only if what Karl Pearson meant by "minor" is what we today would call a "cofactor."

Question

In 1900, did the word "minor" refer to what we would call a "cofactor"?

Terminology: matrix minors and cofactors (involves Cramer's rule)

Background

I'm reading Karl Pearson's 1900 paper titled:

On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling

I paraphrase the opening. He supposes $x_1, x_2, \ldots, x_n$ is a system of random variables following a multivariate normal distribution centered at the origin. Then he asserts the probability density of the vector $\langle x_1, x_2, \ldots, x_n \rangle$ is $$ Z = Z_0 \exp\left( -\frac{1}{2\det S}\sum_i \sum_j S_{pq} x_p x_q\right)\,, $$ where

  • $S$ is the covariance matrix of $x_1, x_2, \ldots, x_n$
  • $S_{pq}$ is the "minor obtained by striking out the $p$th row and $q$th column" of the covariance matrix
  • $Z_0$ is an unspecified constant.

This appears inconsistent with the multivariate normal density function from Wikipedia, which involves a matrix inverse. These two expressions for the density function can be reconciled using Cramer's rule, but only if what Karl Pearson meant by "minor" is what we today would call a "cofactor."

Question

In 1900, did the word "minor" refer to what we would call a "cofactor"?

edited body
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Kimmel
Kimmel
  • 41
  • 5

Background

I'm reading Karl Pearson's 1900 paper titled:

On the Criterion that a given System of Deviations from the ProbablyProbable in the Case of a Correlated System of Variables is such that it can be reasonably supposed to have arisen from Random Sampling

I paraphrase the opening. He supposes $x_1, x_2, \ldots, x_n$ is a system of random variables following a multivariate normal distribution centered at the origin. Then he asserts the probability density of the vector $\langle x_1, x_2, \ldots, x_n \rangle$ is $$ Z = Z_0 \exp\left( -\frac{1}{2\det S}\sum_i \sum_j S_{pq} x_p x_q\right)\,, $$ where

  • $S$ is the covariance matrix of $x_1, x_2, \ldots, x_n$
  • $S_{pq}$ is the "minor obtained by striking out the $p$th row and $q$th column" of the covariance matrix
  • $Z_0$ is an unspecified constant.

This appears inconsistent with the multivariate normal density function from Wikipedia, which involves a matrix inverse. These two expressions for the density function can be reconciled using Cramer's rule, but only if what Karl Pearson meant by "minor" is what we today would call a "cofactor."

Question

In 1900, did the word "minor" refer to what we would call a "cofactor"?

Background

I'm reading Karl Pearson's 1900 paper titled:

On the Criterion that a given System of Deviations from the Probably in the Case of a Correlated System of Variables is such that it can be reasonably supposed to have arisen from Random Sampling

I paraphrase the opening. He supposes $x_1, x_2, \ldots, x_n$ is a system of random variables following a multivariate normal distribution centered at the origin. Then he asserts the probability density of the vector $\langle x_1, x_2, \ldots, x_n \rangle$ is $$ Z = Z_0 \exp\left( -\frac{1}{2\det S}\sum_i \sum_j S_{pq} x_p x_q\right)\,, $$ where

  • $S$ is the covariance matrix of $x_1, x_2, \ldots, x_n$
  • $S_{pq}$ is the "minor obtained by striking out the $p$th row and $q$th column" of the covariance matrix
  • $Z_0$ is an unspecified constant.

This appears inconsistent with the multivariate normal density function from Wikipedia, which involves a matrix inverse. These two expressions for the density function can be reconciled using Cramer's rule, but only if what Karl Pearson meant by "minor" is what we today would call a "cofactor."

Question

In 1900, did the word "minor" refer to what we would call a "cofactor"?

Background

I'm reading Karl Pearson's 1900 paper titled:

On the Criterion that a given System of Deviations from the Probable in the Case of a Correlated System of Variables is such that it can be reasonably supposed to have arisen from Random Sampling

I paraphrase the opening. He supposes $x_1, x_2, \ldots, x_n$ is a system of random variables following a multivariate normal distribution centered at the origin. Then he asserts the probability density of the vector $\langle x_1, x_2, \ldots, x_n \rangle$ is $$ Z = Z_0 \exp\left( -\frac{1}{2\det S}\sum_i \sum_j S_{pq} x_p x_q\right)\,, $$ where

  • $S$ is the covariance matrix of $x_1, x_2, \ldots, x_n$
  • $S_{pq}$ is the "minor obtained by striking out the $p$th row and $q$th column" of the covariance matrix
  • $Z_0$ is an unspecified constant.

This appears inconsistent with the multivariate normal density function from Wikipedia, which involves a matrix inverse. These two expressions for the density function can be reconciled using Cramer's rule, but only if what Karl Pearson meant by "minor" is what we today would call a "cofactor."

Question

In 1900, did the word "minor" refer to what we would call a "cofactor"?

added history tag because the primary question is about how words are historically used (vs today). Removed determinants tag because of weird rule that there can only be five tags
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Vincent
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Kimmel
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