Recipe

Run PCA via the SVD directly

Use when: you need principal components or explained variance for a centered data matrix $X$ , without ever forming the covariance matrix

Goal: the principal directions and the variance each one explains, computed entirely from $X$ 's own singular values

Taught in PCA is an SVD

Quick recipe

Center the data and compute its SVD directly
Read the principal directions off $V$ 's columns directly
Convert each singular value into an explained variance
Compute the proportion of variance each component explains
Confirm the numerical advantage, if it is in question

1
Subtract each variable's mean so $X$ 's columns average to zero, then compute $X=U\Sigma V^{\mathsf T}$ directly; never form $X^{\mathsf T}X$ .

Example

A centered data matrix $X$ with $n=5$ observations has singular values $\sigma_1=8,\sigma_2=4$ .

Why: the covariance-matrix route and the direct SVD route only agree once $X$ is genuinely centered, and skipping straight to the SVD avoids ever squaring $X$ 's own condition number.
2
The columns of $V$ , the right singular vectors, are the principal directions, in the same decreasing order as the singular values, with no separate eigendecomposition to run.

Example

In the point cloud behind this section's covariance-lab widget, $V$ 's first column points along the direction of greatest spread, exactly the widget's own first principal axis.

Why: $C=\frac1{n-1}V\Sigma^2V^{\mathsf T}$ is already $C$ 's own eigendecomposition, and $V$ is already available from Step 1.
3
Divide the squared singular value by $n-1$ : $\lambda_i=\sigma_i^2/(n-1)$ , giving the variance the corresponding principal direction explains.

Example

With $\sigma_1=8,\sigma_2=4,n=5$ : $\lambda_1=64/4=16$ , $\lambda_2=16/4=4$ .

Why: this is exactly the eigenvalue the covariance matrix would have produced, reached without ever computing that matrix.
4
Divide one $\lambda_i$ by the sum of every $\lambda_i$ to get that component's own share of the total variance.

Example

$16/(16+4)=0.8$ , so the first component explains $80\%$ of the total variance.

Why: a raw variance number is hard to interpret alone; the proportion shows whether one or two components already capture most of the data's structure.
5
If $X$ 's own condition number $\kappa(X)$ is known, compare it against $\kappa(X^{\mathsf T}X)=\kappa(X)^2$ to see how much worse the covariance route would have been.

Example

$\kappa(X)=1000$ gives $\kappa(X^{\mathsf T}X)=1{,}000{,}000$ , a million-fold worse conditioning avoided entirely by Step 1's direct route.

Why: this is the concrete numerical payoff for skipping the covariance matrix, not just a theoretical preference.

Common mistakes

Running the SVD on an un-centered data matrix; the link between $\sigma_i$ and explained variance only holds once $X$ is centered.
Dividing by $n$ instead of $n-1$ when converting a singular value into a sample variance.
Forming $X^{\mathsf T}X$ first out of habit, then squaring an already poor condition number for no benefit.
Reading $U$ 's columns as the principal directions instead of $V$ 's; the two play different roles in this identity.