Singular Values Calculator

Your details

Choose the dimensions of your square matrix.
Entry in row 1, column 1.
Entry in row 1, column 2.
Entry in row 2, column 1.
Entry in row 2, column 2.
Singular value 1 (sigma-1)Full rank and well-conditioned
5

Largest singular value (spectral norm of A)

Singular value 2 (sigma-2)1
U matrix (left singular vectors)[[-0.7071, 0.7071], [-0.7071, -0.7071]]
Sigma matrix[[5, 0], [0, 1]]
V-transpose matrix[[-0.7071, -0.7071], [0.7071, -0.7071]]
Matrix rank2
Condition number (kappa)5
Frobenius norm5.09902
Nuclear norm6
sigma-15
sigma-21
sigma-3-

2x2 matrix with full rank: largest singular value 5, condition number 5.

  • The singular values are sigma-1 = 5, sigma-2 = 1. The largest (sigma-1 = 5) is the spectral norm: the maximum factor by which A can stretch any vector.
  • The matrix has full rank (2), meaning it is invertible and maps every direction to a unique output direction.
  • The condition number kappa = 5 is small, so the matrix is well-conditioned: linear systems A x = b solved with this matrix lose very few significant digits.
  • The Frobenius norm is 5.099 (square root of the sum of squared singular values) and the nuclear norm is 6 (sum of all singular values).

Next stepThe full SVD gives you A = U Sigma V-transpose. You can verify it by multiplying the three factor matrices together and checking that the result matches your original A.

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