Matrix Norm Calculator

Enter your matrix values into the grid and select a matrix size (2×2 or 3×3) to compute five key matrix norms: the 1-norm (max column sum), infinity-norm (max row sum), Frobenius norm, max-norm, and 2-norm (largest singular value approximation). Results update automatically as you change your matrix entries.

Choose the dimensions of your matrix.

Results

Frobenius Norm ‖A‖_F

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1-Norm ‖A‖₁ (Max Column Sum)

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Infinity-Norm ‖A‖∞ (Max Row Sum)

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Max-Norm ‖A‖_max (Largest Entry)

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2-Norm ‖A‖₂ (Spectral Norm Approx.)

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Matrix Norm Comparison

Frequently Asked Questions

What is the norm of a matrix?

A matrix norm is a scalar value that measures the 'size' or 'magnitude' of a matrix, much like the length of a vector. Unlike a vector norm, a matrix norm often represents the maximum stretching factor a matrix can apply to a unit vector. Different norm definitions capture different aspects of a matrix's structure.

What is the 1-norm of a matrix?

The 1-norm (also called the column-sum norm) is computed by summing the absolute values of each column's entries, then taking the maximum across all columns. Formally, ‖A‖₁ = max_j Σᵢ |aᵢⱼ|. It measures the largest total absolute column weight.

What is the Frobenius norm of a matrix?

The Frobenius norm is the square root of the sum of the squares of all matrix entries: ‖A‖_F = √(Σᵢ Σⱼ |aᵢⱼ|²). It is analogous to the Euclidean (L2) norm for vectors and is one of the most commonly used matrix norms in numerical analysis and machine learning.

What is the Frobenius norm of the identity matrix?

For an n×n identity matrix, the Frobenius norm equals √n. For a 2×2 identity matrix it is √2 ≈ 1.4142, and for a 3×3 identity matrix it is √3 ≈ 1.7321, since the diagonal entries are all 1 and all other entries are 0.

Do rectangular matrices have norms?

Yes. Matrix norms such as the 1-norm, infinity-norm, Frobenius norm, and max-norm are all defined for any m×n matrix, not just square matrices. The 2-norm (spectral norm) also extends to rectangular matrices as the largest singular value.

What does ‖A‖ mean in matrices?

The notation ‖A‖ denotes a matrix norm of matrix A. Without a subscript it typically refers to the induced 2-norm (spectral norm), which equals the largest singular value of A. With subscripts like ‖A‖₁, ‖A‖_F, or ‖A‖∞, it refers to the 1-norm, Frobenius norm, or infinity-norm respectively.

Can a matrix norm be less than 1?

Yes. A matrix norm can be any non-negative real number, including values less than 1. For example, a matrix with very small entries will have small norms. A norm equals zero only for the zero matrix.

What is the difference between the 2-norm and the Frobenius norm?

The 2-norm (spectral norm) equals the largest singular value of the matrix and measures maximum stretching. The Frobenius norm sums all squared entries and is generally larger than or equal to the 2-norm. They coincide for rank-1 matrices. Both satisfy the standard norm axioms.

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