SVD Calculator - Singular Value Decomposition

Your details

Choose whether to decompose a 2x2 or 3x3 real matrix.
Number of decimal places shown in U, Sigma, and V^T.
First row of the 2x2 matrix. Separate values with commas or spaces.
Second row of the 2x2 matrix.
Singular value s1
5.1167

Largest singular value of A (square root of the largest eigenvalue of A^T*A)

Singular value s21.9544
Rank2
Condition number2.62
Matrix U[0.5257, 0.8507] [0.8507, -0.5257]
Matrix Sigma[5.1167, 0.0000] [0.0000, 1.9544]
Matrix V^T[0.6407, 0.7678] [0.7678, -0.6407]
Verification error0
s1 (largest)5.1167
s21.9544
s3 (3x3 only)-

SVD complete: 2x2 matrix decomposed into U, Sigma, V^T.

  • The largest singular value is 5.1167, representing the maximum "stretching" direction of the transformation.
  • The second singular value is 1.9544. The ratio s1/s2 = 2.62 shows how much more dominant the first direction is.
  • The matrix has full rank 2, meaning it is invertible and maps no non-zero vector to zero.
  • The condition number is 2.62, so the matrix is well-conditioned (low sensitivity to numerical errors).

Next stepTo use the SVD for dimensionality reduction, keep only the k largest singular values and their corresponding columns of U and rows of V^T. This gives the best rank-k approximation of A.

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