How this calculator works
Matrix norms measure the size of a matrix. The 1-norm adds absolute values in each column and takes the largest sum. The infinity-norm does the same by rows. The 2-norm is the largest singular value, which is the maximum stretching factor on a unit vector. The Frobenius norm is the square root of the sum of the squares of all entries. The entrywise max norm is the largest absolute entry. The first three are induced operator norms; the last two are convenient entrywise measures.
Formula
||A||1 = max column sum of |aij|; ||A||inf = max row sum of |aij|; ||A||2 = largest singular value; ||A||F = sqrt(sum of aij^2); ||A||max = max |aij|.
Worked example
The matrix with rows 1, 2 and 3, 4 has column sums 4 and 6, so its 1-norm is 6. Its row sums are 3 and 7, so its infinity-norm is 7. Its Frobenius norm is the square root of 30, about 5.477, its largest entry is 4 and its largest singular value is about 5.465.
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Frequently asked questions
Which norm should I use?
Use the 2-norm for geometric questions about stretching, the 1-norm or infinity-norm for fast bounds, and the Frobenius norm when you want a simple total of the entry sizes.
Are all matrix norms equivalent?
For finite matrices they are equivalent up to constant factors that depend on the dimensions, so they agree on convergence even though their numeric values differ.
Why is the 2-norm a singular value?
The singular value decomposition shows that the matrix maps perpendicular directions by the singular values. The largest one is therefore the maximum length of A x for a unit vector x.