Singular Values Calculator

Enter your matrix values into the Singular Values Calculator and get the singular values (σ₁, σ₂, σ₃) of any matrix up to 3×3. Input your matrix dimensions (rows and columns) and fill in the matrix entries — the tool computes ATA, finds its eigenvalues, and returns the sorted singular values as square roots of those eigenvalues. Great for linear algebra, data science, and SVD decomposition work.

Select the number of rows × columns of your matrix.

Used only for matrices with 3 columns.

Used only for matrices with 3 columns.

Used only for matrices with 3 rows.

Used only for matrices with 3 rows.

Used only for 3×3 matrices.

Results

Largest Singular Value (σ₁)

--

Second Singular Value (σ₂)

--

Third Singular Value (σ₃)

--

Matrix Rank (non-zero σ)

--

Frobenius Norm (√Σσ²)

--

Singular Values Distribution

Results Table

Frequently Asked Questions

What are the singular values of a matrix?

The singular values of a matrix A are the square roots of the eigenvalues of AᵀA (or AA* for complex matrices). They are always non-negative real numbers, typically listed in decreasing order (σ₁ ≥ σ₂ ≥ … ≥ 0). They appear in the Singular Value Decomposition (SVD): A = UΣVᵀ, where Σ is a diagonal matrix of singular values.

How do I find the singular values of a matrix?

To find singular values: (1) Compute AᵀA (the transpose of A multiplied by A). (2) Find the eigenvalues λ₁, λ₂, … of AᵀA. (3) Take the square root of each non-negative eigenvalue — these are the singular values σᵢ = √λᵢ. This calculator automates all three steps for you.

Are singular values the same as eigenvalues?

No. Eigenvalues are defined for square matrices and can be negative or complex, while singular values are always non-negative real numbers and exist for any m×n matrix. For symmetric positive semi-definite matrices, the singular values equal the eigenvalues. In general, singular values are the square roots of the eigenvalues of AᵀA, not of A itself.

What are the singular values of a diagonal matrix?

For a diagonal matrix, the singular values are simply the absolute values of the diagonal entries, sorted in descending order. For example, if A = diag(3, −5), then the singular values are σ₁ = 5 and σ₂ = 3.

What are the singular values of a symmetric matrix?

For a real symmetric matrix, the singular values are the absolute values of the eigenvalues of that matrix, sorted in descending order. This is because AᵀA = A² for symmetric A, so the eigenvalues of AᵀA are the squares of the eigenvalues of A.

What does the rank of a matrix have to do with singular values?

The rank of a matrix equals the number of its non-zero singular values. If a matrix has r non-zero singular values, it has rank r. This makes singular values a numerically stable way to determine rank, especially for nearly-singular matrices where direct methods may give inaccurate results.

What is the Frobenius norm and how does it relate to singular values?

The Frobenius norm of a matrix is the square root of the sum of all squared entries. It equals the square root of the sum of the squares of all singular values: ‖A‖_F = √(σ₁² + σ₂² + … + σₙ²). It is a measure of the overall magnitude of the matrix.

Can this calculator handle non-square matrices?

Yes. Singular values are defined for any m×n matrix, whether square or rectangular. This calculator supports 2×2, 2×3, 3×2, and 3×3 matrices. For a 2×3 matrix, AᵀA is a 3×3 matrix and you get up to 3 singular values; for a 3×2 matrix, AᵀA is 2×2 and you get up to 2 singular values.

More Math Tools