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PCA projects centered observations onto covariance eigenvectors ordered by explained variance.
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Formula
Eigenvalues have squared feature units; eigenvectors are unitless; scores use feature units
| Symbol | Meaning | Unit |
|---|---|---|
| Feature vector of an observation | feature units | |
| x̄ | Mean vector of the training data | feature units |
| Sample covariance matrix of centered training data | squared feature units | |
| Normalized eigenvector for principal direction k | dimensionless | |
| Variance explained along direction k | squared feature units | |
| Coordinate of the centered observation along direction k | trait unit |
S v_k=\lambda_k v_k,\quad z_k=(x-\bar{x})^T v_kFor a symmetric covariance matrix, principal directions are orthonormal eigenvectors ordered by descending eigenvalue. Center features first; standardize only when comparable scale is appropriate, because that changes the covariance problem. The sign of an eigenvector is arbitrary. PCA is linear and unsupervised; high variance is not automatically predictive or causally meaningful. With correlated or differently scaled features, state the preprocessing and covariance convention.
For a symmetric covariance matrix, principal directions are orthonormal eigenvectors ordered by descending eigenvalue. Center features first; standardize only when comparable scale is appropriate, because that changes the covariance problem. The sign of an eigenvector is arbitrary. PCA is linear and unsupervised; high variance is not automatically predictive or causally meaningful. With correlated or differently scaled features, state the preprocessing and covariance convention.
PCA projects centered observations onto covariance eigenvectors ordered by explained variance.
Defining relation
The equation states the defining relation within its stated domain.
Use the example for Principal component analysis (PCA) to explain which inputs determine the requested output.
The example connects concrete inputs to its result.
PCA projects centered observations onto covariance eigenvectors ordered by explained variance.
The relation applies only under its stated assumptions.
Applying the relation to a different case without checking its stated assumptions.
Check the validity range before substituting values: For a symmetric covariance matrix, principal directions are orthonormal eigenvectors ordered by descending eigenvalue. Center features first; standardize only when comparable scale is appropriate, because that changes the covariance problem..
Formula cluster
PCA projects centered observations onto covariance eigenvectors ordered by explained variance.
For points (2,0) and (0,2), the mean is (1,1) and the sample covariance is [[2,−2],[−2,2]]. The leading unit eigenvector is (1,−1)/√2 with eigenvalue 4; projections of the centered points are ±√2. The orthogonal component has zero variance.
Reduce feature dimension and summarize dominant directions of variation.
Here is how to work through a typical Principal component analysis (PCA) (Eigenvektoren der Kovarianzmatrix; Projektion zₖ) task step by step:
Task
Use the example for Principal component analysis (PCA) to explain which inputs determine the requested output.
Solution path
The example connects concrete inputs to its result.
PCA projects centered observations onto covariance eigenvectors ordered by explained variance.
The relation applies only under its stated assumptions.
The relation S vk=\lambda_k vk,\quad zk=(x-\bar{x})^T vk summarizes PCA projects centered observations onto covariance eigenvectors ordered by explained variance. Read it as a model for defined quantities, not as an automatic claim about every real situation. In the example For points (2,0) and (0,2), the mean is (1,1) and the sample covariance is [[2,−2],[−2,2]]. The leading unit eigenvector is (1,−1)/√2 with eigenvalue 4; projections of the centered points are ±√2. The orthogonal component has zero variance., symbols are connected to concrete values and the result is interpreted in context. Validity limit: For a symmetric covariance matrix, principal directions are orthonormal eigenvectors ordered by descending eigenvalue. Center features first; standardize only when comparable scale is appropriate, because that changes the covariance problem. Source: scikit-learn User Guide, §2.5.1.1 Exact PCA and probabilistic interpretation.
First write down the requested quantity and all given values with their symbols. Substitute them into the defining relation and check unit consistency. The example For points (2,0) and (0,2), the mean is (1,1) and the sample covariance is [[2,−2],[−2,2]]. The leading unit eigenvector is (1,−1)/√2 with eigenvalue 4; projections of the centered points are ±√2. The orthogonal component has zero variance. provides a worked check: formula and result must have compatible scale and meaning; round only at the end. Validity limit: For a symmetric covariance matrix, principal directions are orthonormal eigenvectors ordered by descending eigenvalue. Center features first; standardize only when comparable scale is appropriate, because that changes the covariance problem. Source: scikit-learn User Guide, §2.5.1.1 Exact PCA and probabilistic interpretation.
The formula applies only when quantities are defined as in the source and background. Check boundary cases, measurement conditions and whether an approximation rather than equality is intended. Substitution can be algebraically correct yet technically wrong when assumptions fail or the population or system boundary changes. Validity limit: For a symmetric covariance matrix, principal directions are orthonormal eigenvectors ordered by descending eigenvalue. Center features first; standardize only when comparable scale is appropriate, because that changes the covariance problem. Source: scikit-learn User Guide, §2.5.1.1 Exact PCA and probabilistic interpretation.
The stated units are: Eigenvalues have squared feature units; eigenvectors are unitless; scores use feature units. Convert values to a consistent unit system before substitution; counts, percentages and dimensionless ratios are not interchangeable. The variable list states the role of each symbol. Dimensional analysis catches many errors but does not replace checking the formula's assumptions. Validity limit: For a symmetric covariance matrix, principal directions are orthonormal eigenvectors ordered by descending eigenvalue. Center features first; standardize only when comparable scale is appropriate, because that changes the covariance problem. Source: scikit-learn User Guide, §2.5.1.1 Exact PCA and probabilistic interpretation.
Comparison helps when both quantities address the same question but emphasize different properties. Report the definition, input data, threshold or averaging with the value. The source scikit-learn User Guide, §2.5.1.1 Exact PCA and probabilistic interpretation. supports the definition used here; other conventions can differ. Choose the measure to match the learning or analysis goal. Validity limit: For a symmetric covariance matrix, principal directions are orthonormal eigenvectors ordered by descending eigenvalue. Center features first; standardize only when comparable scale is appropriate, because that changes the covariance problem. Source: scikit-learn User Guide, §2.5.1.1 Exact PCA and probabilistic interpretation.
Quanta exam set
Curated exam set for "Principal component analysis (PCA)":
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Which formula describes Principal component analysis (PCA)?
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Question (front)
How do you rearrange Principal component analysis (PCA) for Defining relation?
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Question (front)
Which common mistake happens with Principal component analysis (PCA)?
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Create a curated FSRS exam set for Eigenvektoren der Kovarianzmatrix; Projektion zₖ: formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.
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