Matrix norms measure the “size” of a matrix. They bound errors in numerical algorithms, define condition numbers and measure distances between matrices in data science.
How to use the Matrix Norm Calculator
- Choose the matrix size – the matching grid of fields appears.
- Enter the entries row by row (a₁ b₁ c₁ is the first row, a₂ b₂ c₂ the second, and so on).
- Read the result; integer matrices give exact fractions.
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
The column and row sums give the 1- and ∞-norms directly. The 2-norm is the largest singular value, found from the eigenvalues of AᵀA. The Frobenius norm is the Euclidean length of all entries.
Formula
‖A‖₁ = max column sum of |aᵢⱼ|, ‖A‖∞ = max row sum, ‖A‖₂ = largest singular value
‖A‖_F = √(Σ aᵢⱼ²), ‖A‖max = max |aᵢⱼ|
Worked example
Inputs — Matrix size (rows × columns): 2 × 2; a₁: 1; b₁: -2; a₂: 3; b₂: 4.
Results — 2-norm (spectral) = 5.116672736; 1-norm = 6; ∞-norm = 7; Frobenius norm = 5.477225575; Max norm = 4.
Key terms
- Matrix norm
- A measure of the size of a matrix.
Frequently asked questions
Which matrix norm should I use?
The 2-norm for theory, the 1- or ∞-norm for quick bounds, Frobenius for least squares.
Is the max norm submultiplicative?
No – ‖AB‖ ≤ ‖A‖‖B‖ can fail for it.
How are the norms related?
‖A‖₂ ≤ ‖A‖_F ≤ √r ‖A‖₂ for rank r.
What is an induced norm?
A matrix norm defined from a vector norm, such as the 1-, 2- and ∞-norms.