Singular Values Calculator
Enter a real matrix and calculate its singular values in descending order, with a ready-to-export Excel workbook.
Matrix inputs
Accepted examples include -12, 3.5, and 1,250.25. Decimal commas and scientific notation are rejected to prevent ambiguous input.
Singular values
The calculator forms a symmetric Gram matrix, finds its non-negative eigenvalues, and takes their square roots: σᵢ = √λᵢ.
Calculation detail
| Order | Singular value (σ) | Squared value (λ) |
|---|---|---|
| 1 | 6.70820393 | 45 |
| 2 | 2.23606798 | 5 |
The squared values are the eigenvalues of AᵀA or AAᵀ, depending on which produces the smaller symmetric matrix. Values are sorted from largest to smallest.
How to use the singular values calculator
What this calculator does
This calculator finds the singular values of a real rectangular or square matrix with one to six rows and one to six columns. Singular values describe how strongly a matrix stretches vectors along special orthogonal directions. They are always real, non-negative, and listed from largest to smallest. The tool estimates the values numerically; it does not return the left and right singular vectors or a full symbolic singular value decomposition.
When to use it
Use it to check linear-algebra homework, compare the strength of different matrix directions, identify whether a small matrix is numerically rank deficient, or verify values before using a matrix in least-squares, compression, dimensionality-reduction, or conditioning work. For a deeper geometric explanation of the factorization A = UΣVᵀ, see the MIT notes on the singular value decomposition.
How to calculate
- The calculator opens with a complete 2 × 2 demonstration matrix, so the results and the example Excel workbook are available immediately.
- Choose the Number of rows and Number of columns. The matrix-entry grid changes to match those dimensions.
- Replace every Matrix entries value. Results update live after each valid edit. Read the largest value first, then review the complete ordered table.
- Select Download Excel to create a current-state workbook containing the matrix, summary outputs, and full singular-value table.
- Select Reset to clear the demonstration values. Reset keeps a neutral 2 × 2 shape but empties every entry, clears results, and disables Download Excel until a complete valid matrix is entered again.
Input guide
Number of rows is a required whole-number selector from 1 through 6; the startup example uses 2. Increasing it adds another horizontal matrix dimension and can increase the number of independent directions, although the number of singular values can never exceed the smaller matrix dimension. Number of columns is also a required 1-to-6 selector and starts at 2. Changing either selector preserves entries that still occupy the same row-and-column position; newly created positions are blank and must be completed. Do not confuse rows with columns when transcribing a matrix.
Matrix entries are required real numbers. Enter plain U.S.-style decimals such as 4, -2.5, or 1,250.25. Proper thousands separators are accepted, but decimal-comma input such as 1,5 and scientific notation such as 1e3 are rejected rather than silently reinterpreted. Zero and negative values are valid. Each entry affects the Gram matrix and may change every singular value, so a one-cell change can alter both the largest value and the numerical rank.
Output guide
The Matrix pill reports the active dimensions, while Values reports min(rows, columns), the exact count of singular values returned. The Rank pill and Numerical rank result count values larger than a floating-point tolerance tied to matrix size and the largest singular value. A rank below min(rows, columns) indicates numerical rank deficiency; a zero matrix has rank 0.
Largest singular value is σ₁, also called the spectral or Euclidean operator norm of the matrix. A larger value means the matrix can stretch at least one unit vector more strongly. Smallest singular value is the final value in the returned sequence; zero or a value near zero signals a lost or nearly lost direction. In the Calculation detail table, Order is the descending position, Singular value (σ) is the square root result, and Squared value (λ) is the corresponding eigenvalue of the symmetric Gram matrix. These are numerical estimates, not symbolic radicals.
Worked example
The startup matrix is A = [3, 0], [4, 5][25, 20], [20, 25]