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Matrix Norm Calculator

Enter the four entries of your 2x2 matrix to find its 1-norm and infinity-norm.

How it is calculated

The matrix is written as rows [a b] and [c d]. Each entry is replaced by its absolute value, so signs never matter. The 1-norm adds the absolute values down each column and keeps the larger total: ‖A‖₁ = max(|a|+|c|, |b|+|d|). The infinity-norm adds across each row and keeps the larger total: ‖A‖∞ = max(|a|+|b|, |c|+|d|). The tool also reports the largest single absolute entry, max(|a|,|b|,|c|,|d|). All entries are plain numbers with no units, and results are rounded to two decimals. It does not compute the spectral (2-)norm or the Frobenius norm, which need squares and eigenvalues. A blank or non-numeric entry gives no result.

Worked example

Take the matrix with a = 1, b = −2, c = 3, d = 4. Column sums of absolute values: first column |1| + |3| = 4, second column |−2| + |4| = 6. The larger is 6, so the 1-norm is 6.00. Row sums: first row |1| + |−2| = 3, second row |3| + |4| = 7, so the infinity-norm is 7.00. The largest single absolute entry is 4.00. Notice that the two norms differ for the same matrix, because one looks at columns and the other at rows.

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Common mistakes

  • Adding the entries with their signs. The minus sign on −2 must be dropped first; summing 1 + 3 or −2 + 4 gives the wrong column total.
  • Mixing up the two norms: the 1-norm uses columns, the infinity-norm uses rows. For a non-symmetric matrix the answers differ.
  • Entering the entries in the wrong positions, for example column by column instead of row by row. This swaps the two norms for a non-symmetric matrix.
  • Expecting the result to be the spectral norm (largest singular value). The 1-norm and infinity-norm are only upper bounds on the largest eigenvalue magnitude, not equal to it.

About this calculator

A matrix norm gives a single number that measures how much a matrix can “stretch” a vector, and different norms answer that question in different ways. For a matrix induced by vector p-norms, the 1-norm equals the largest of the absolute column sums (worst-case stretch when the input is measured with the taxicab / 1-norm), while the infinity-norm equals the largest of the absolute row sums (worst-case stretch under the max-norm) — both quick to compute by hand for a small matrix, unlike the spectral (2-)norm, which needs eigenvalues.

Numerical analysts and engineers use these norms to bound error propagation in linear systems, estimate the condition number of a matrix (how sensitive a solution is to small input errors), and check convergence criteria in iterative solvers. From the four entries of a 2×2 matrix it reads off the 1-norm (largest absolute column sum), the infinity norm (largest absolute row sum) and the largest absolute entry.

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  • Editorial team

    A quick use is error bounding: the infinity-norm and 1-norm each give an upper limit on how much a 2 by 2 matrix can magnify an input, so the smaller of the two is the tighter bound you can state. For the matrix with entries 1, -2, 3, 4 they are 6.00 and 7.00. Transposing the matrix swaps them, which is a quick way to confirm your entries sit in the right positions. Which norm do you need that is missing here?

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