SVD and Engineering Computation
Cheat sheet.

The key rules, formulas and reminders from all 3 topics, gathered into reference cards.

Key formulas, conditions and traps · Read down each column.

Singular Value Decomposition

Core rule

A=U\Sigma V^T,\quad Av_i=\sigma_i u_i,\quad \sigma_i^2\text{ are eigenvalues of }A^TA.

Watch for

Singular values are nonnegative and exist for rectangular matrices. Do not divide by a zero singular value.

Pseudoinverses, Low Rank and Conditioning

Core rule

A^+=V\Sigma^+U^T,\quad x_*=A^+b,\quad \kappa_2(A)=\sigma_{\max}/\sigma_{\min}\text{ for invertible }A.

Watch for

Small singular values amplify noise. Truncation trades accuracy in the original model for reduced sensitivity.

Linear Algebra Checkpoint

Core rule

Use elimination for exact systems, QR for full-column-rank least squares, eigenvectors for invariant directions, and SVD for general rectangular geometry and sensitivity.

Watch for

Report whether a solution exists and is unique. Separate residual error, measurement sensitivity and numerical rounding.

Singular Value Decomposition

2 reference blocks

Read lesson ↗

Core rule

A=U\Sigma V^T,\quad Av_i=\sigma_i u_i,\quad \sigma_i^2\text{ are eigenvalues of }A^TA.

Watch for

Singular values are nonnegative and exist for rectangular matrices. Do not divide by a zero singular value.

Pseudoinverses, Low Rank and Conditioning

2 reference blocks

Read lesson ↗

Core rule

A^+=V\Sigma^+U^T,\quad x_*=A^+b,\quad \kappa_2(A)=\sigma_{\max}/\sigma_{\min}\text{ for invertible }A.

Watch for

Small singular values amplify noise. Truncation trades accuracy in the original model for reduced sensitivity.

Linear Algebra Checkpoint

2 reference blocks

Read lesson ↗

Core rule

Use elimination for exact systems, QR for full-column-rank least squares, eigenvectors for invariant directions, and SVD for general rectangular geometry and sensitivity.

Watch for

Report whether a solution exists and is unique. Separate residual error, measurement sensitivity and numerical rounding.