Advanced SVD Calculator with Steps

Solve decompositions fast with guided matrix breakdowns. View orthogonal factors, singular spectrum, and exact reconstruction. Practice linear algebra concepts using intuitive exports and visuals.

Enter matrix data

Use one row per line. Separate values with spaces or commas.

Example data table

This sample matrix works well for testing steps, exports, and the graph.

	Column 1	Column 2
Row 1	3	1
Row 2	2	2
Row 3	0	1

Sample entry format: 3 1, 2 2, and 0 1 on separate lines.

Formula used

Core identity: A = UΣVᵀ

Symmetric step: AᵀA = VΛVᵀ

Singular values: σᵢ = √λᵢ

Left vectors: uᵢ = Avᵢ / σᵢ, when σᵢ > 0

Check: UΣVᵀ should reconstruct the original matrix with tiny numerical error.

How to use this calculator

Enter the matrix size first.
Paste one matrix row on each line.
Separate numbers using spaces or commas.
Adjust precision, tolerance, and iteration settings.
Press Calculate SVD to generate steps.
Review U, Σ, Vᵀ, spectrum, and reconstruction.
Use CSV or PDF buttons to export results.
Load the sample matrix if you want a quick test.

Frequently asked questions

1. What does SVD mean?

SVD means singular value decomposition. It factors a matrix into left singular vectors, singular values, and right singular vectors. This reveals structure, rank, and directional strength.

2. Why does the calculator form AᵀA first?

AᵀA is symmetric and easier to analyze numerically. Its eigenvalues are nonnegative, and their square roots become the singular values used in the decomposition.

3. What does the rank value show?

The rank estimates how many singular values are meaningfully nonzero. It tells you how many independent directions exist in the matrix.

4. Why can small reconstruction errors appear?

Tiny errors come from rounding and iterative numerical computation. A very small difference matrix usually confirms the decomposition is working correctly.

5. Can this tool handle rectangular matrices?

Yes. The calculator accepts rectangular inputs up to the allowed size limits. The compact decomposition still shows the essential singular structure clearly.

6. What is the condition number used for?

The condition number compares the largest and smallest nonzero singular values. Larger values suggest greater sensitivity to input changes or numerical noise.

7. When should I change tolerance or iterations?

Increase iterations for harder matrices or stricter convergence. Adjust tolerance when you want smaller values treated as numerical zeros during rank estimation.

8. What does the graph show?

The graph plots singular values and cumulative energy. It helps you see dominant components quickly and judge how much information each component captures.