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PrincipalComponents[matrix]

transforms elements of matrix into unscaled principal components.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
Method  
Properties & Relations  
Possible Issues  
Neat Examples  
See Also
Related Guides
History
Cite this Page

PrincipalComponents

PrincipalComponents[matrix]

transforms elements of matrix into unscaled principal components.

Details and Options

  • PrincipalComponents gives the principal component transform of matrix.
  • The principal components of matrix are linear transformations of the original columns into uncorrelated columns arranged in order of decreasing variance.
  • PrincipalComponents supports a Method option. The following explicit settings can be specified:
  • "Covariance"uses covariance method (default)
    "Correlation"uses correlation method
  • If principal components of scaled columns (standardized principal components) are required, the option Method"Correlation" should be used.
  • The dimensions of PrincipalComponents[matrix] are the same as the dimensions of matrix.
  • If matrix consists of exact numbers or symbols, the result is also exact or symbolic, respectively.

Examples

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Basic Examples  (1)

Principal components of two datasets, treated as the columns of a matrix:

Wolfram Language code: PrincipalComponents[N[{{1, 2}, {2, 3}, {4, 10}}]]

Scope  (3)

Principal components computed with arbitrary-precision numbers:

Wolfram Language code: PrincipalComponents[{{1`20, 2`20}, {3`20, 5`20}, {5`20, 6`20}}]

Principal components of exact numbers:

Wolfram Language code: PrincipalComponents[{{1, 3}, {3, 5}, {4, 5}}]//Simplify

Principal components computation involving symbolic expressions:

Wolfram Language code: PrincipalComponents[{{a, 1}, {2, a - 1}}]//FullSimplify

Options  (1)

Method  (1)

Principal components using correlation scaling:

Wolfram Language code: PrincipalComponents[{{1., 2., -1.}, {2., 3., 2.}, {4., 10., 9.}, {4., 5., 6.}}, Method -> "Correlation"]

Properties & Relations  (2)

The principal component columns are ordered by decreasing variance:

Wolfram Language code: data = {{13.2, 200, 58, 21.2}, {10, 263, 48, 44.5}, {8.1, 294, 80, 31}, {8.8, 190, 50, 19.5}, {9, 276, 91, 40.6}, {7.9, 204, 78, 38.7}, {3.3, 110, 77, 11.1}, {5.9, 238, 72, 15.8}, {15.4, 335, 80, 31.9}, {17.4, 211, 60, 25.8}};
Wolfram Language code: Variance[pc = PrincipalComponents[data]]

The mean of each principal component column is zero:

Wolfram Language code: Mean[pc] // Chop

The principal component columns are not correlated:

Wolfram Language code: Correlation[pc]//Chop

The setting Method->"Correlation" yields the same results as standardizing the input matrix:

Wolfram Language code: PrincipalComponents[{{1., 2., -1.}, {2., 3., 2.}, {4., 10., 9.}}, Method -> "Correlation"] == PrincipalComponents[Standardize[{{1., 2., -1.}, {2., 3., 2.}, {4., 10., 9.}}]]

Possible Issues  (1)

For certain symbolic matrices the result may be very large:

Wolfram Language code: PrincipalComponents[{{a, 1}, {1, b}, {2, c}}] // LeafCount

Neat Examples  (1)

Align the principal axis of a two-dimensional shape with the horizontal axis:

Wolfram Language code: shape = Position[ImageData@[image], 1, {2}]; ListPlot[PrincipalComponents[N@shape]]
Wolfram Research (2010), PrincipalComponents, Wolfram Language function, https://reference.wolfram.com/language/ref/PrincipalComponents.html.

Text

Wolfram Research (2010), PrincipalComponents, Wolfram Language function, https://reference.wolfram.com/language/ref/PrincipalComponents.html.

CMS

Wolfram Language. 2010. "PrincipalComponents." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PrincipalComponents.html.

APA

Wolfram Language. (2010). PrincipalComponents. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PrincipalComponents.html

BibTeX

@misc{reference.wolfram_2026_principalcomponents, author="Wolfram Research", title="{PrincipalComponents}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/PrincipalComponents.html}", note=[Accessed: 15-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_principalcomponents, organization={Wolfram Research}, title={PrincipalComponents}, year={2010}, url={https://reference.wolfram.com/language/ref/PrincipalComponents.html}, note=[Accessed: 15-June-2026]}