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(linear algebra) A particular type of factorisation of a matrix into a product of three matrices, of which the second is a diagonal matrix that has as the entries on its diagonal the singular values of the original matrix.

Illustration of the singular value decomposition UΣV^⁎ of a real 2 × 2 matrix M. Top: The action of M, indicated by its effect on the unit disc D and the two canonical unit vectors e1 and e2. Left: The action of V⁎, a rotation, on D, e1, and e2. Bottom: The action of Σ, a scaling by the singular values σ1 horizontally and σ2 vertically. Right: The action of U, another rotation.

(linear algebra) A particular type of factorisation of a matrix into a product of three matrices, of which the second is a diagonal matrix that has as the entries on its diagonal the singular values of the original matrix.

Animated illustration of the SVD of a 2D, real shearing matrix M. First, we see the unit disc in blue together with the two canonical unit vectors. We then see the actions of M, which distorts the disk to an ellipse. The SVD decomposes M into three simple transformations: an initial rotation V^⁎, a scaling ${\displaystyle \mathbf {\Sigma } }$ along the coordinate axes, and a final rotation U. The lengths σ₁ and σ₂ of the semi-axes of the ellipse are the singular values of M, namely Σ_1,1 and Σ_2,2.

(linear algebra) A particular type of factorisation of a matrix into a product of three matrices, of which the second is a diagonal matrix that has as the entries on its diagonal the singular values of the original matrix.

Visualization of the matrix multiplications in singular value decomposition