Singular Values of a Matrix Calculator

Calculator Inputs

Formula Used

For an m by n matrix A, singular values are found from the symmetric matrix A^TA.

A^TA v = λv

σ_i = √λ_i

The calculator computes A^TA, estimates its eigenvalues with Jacobi rotations, and returns sorted square roots. It also reports rank, spectral norm, Frobenius norm, nuclear norm, stable rank, and condition number.

How to Use This Calculator

1. Enter the number of rows and columns.
2. Type each matrix row on a new line.
3. Separate values with spaces, commas, or semicolons.
4. Choose decimal places and a numerical tolerance.
5. Press the calculate button to see results above the form.
6. Use CSV or PDF export after a successful calculation.

Example Data Table

Understanding Singular Values

Singular values describe the stretching strength of a matrix. They work for square and rectangular matrices. That makes them useful in many algebra tasks. A large singular value shows a direction with strong scaling. A small singular value shows weak scaling. A zero value shows lost dimension.

Why They Matter

These values help explain rank, stability, compression, and error growth. Engineers use them in signals and control work. Data teams use them in low rank models. Students use them to understand transformations. The largest singular value is the spectral norm. It measures the greatest possible stretch. The sum of all singular values is the nuclear norm. The square root of the squared sum gives the Frobenius norm.

How This Calculator Helps

This calculator builds A transpose A from your entries. It then finds the eigenvalues of that symmetric matrix. Singular values are the square roots of those eigenvalues. The tool sorts the values from largest to smallest. It also estimates rank using your tolerance. The condition number compares the largest value with the smallest reliable value. A high condition number can mean sensitive calculations.

Practical Reading Tips

Do not study one number alone. Compare the spread between values. A flat list means balanced scaling. A list with one dominant value suggests a main direction. Tiny values may be numerical noise. Adjust tolerance when entries are very large or very small. Use more decimals for classroom checking. Use fewer decimals for reports. CSV export supports spreadsheets. PDF export supports clean sharing. Always confirm entries before using results in assessed work.

Common Matrix Uses

Singular values appear in least squares, image compression, principal components, and numerical rank checks. They also reveal whether a matrix is close to losing information. For rectangular matrices, they give a clear size-independent summary. This makes the method stronger than determinant checks. Determinants only apply to square matrices. Singular values apply widely. They show geometry, stability, and structure in one ordered list.

For best accuracy, enter rows consistently. Avoid blank cells inside the matrix. Decimal and negative entries are accepted. Fractions should be converted first. When results seem unexpected, inspect the energy share column and retry with a smaller tolerance before drawing final conclusions.

FAQs