Singular Value Decomposition

If a matrix A stretches and sends an orthonormal basis {u_1, u_2} to an orthonormal basis {v_1, v_2} then you can compute A x as a three step process.

The singular value decomposition of A writes A as a product of three matrices A = M_3M_2M_1 where matrix M_i does step i.

Three steps to compute A x .

Step 1:  Find the u_i coordinates for x .  
Since {u_1, u_2} is an orthonormal basis the u -coords for x are c_iOverscript[=, (1)] u_i · x.  
    (1) See "orthonormal coordinates" in Getting Started.
Choose M_1 = (u ) 1 u 2 to be the matrix with rows u_i.
Then M_1x   = (u ) x Overscript[=, (2)] (u · x ) = (c ) 1 ... u · x c 2 2 2.
    (2) See "row way" in Getting Started.

Step 2:  Multiply the u_i coordinates (c ) 1 c 2 by the singular values s_1 and s_2.
Choose the matrix M_2 = (s ) 1 0 s 0 2.  
Then M_2 (c ) = (s ) (c ) = (s c ) 1 1 0 1 1 1 c s c s c 2 0 2 2 2 2.

Step 3:  Use (s c ) 1 1 s c 2 2 as the v_i coordinates of A x .
Choose M_3 = (v v ) 1 2 to be the matrix with columns v_i.
Then M_3(s c ) = (v v ) (s c ) Overscript[=, (3)] s_1c_1v_1 + s_2c_2v_2 1 1 1 2 1 1 s c s c 2 2 2 2.
    (3) See "column way" in Getting Started.

Combining all three steps you get
    A x = M_3M_2M_1x = (v v ) (s ) (u ) x 1 2 1 0 1 s u 0 2 2.

The singular value decomposition for A is
    A = (v v ) (s ) (u ) 1 2 1 0 1 s u 0 2 2.

Here's the picture of xA x for another 2 × 2 matrix A.

(Q19)  In terms of the vectors {z_1, z_2} and {w_1, w_2}, the singular value decomposition for A is A = (         ) ( &n ...           .

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Copyright © 2007 Todd Will
Created by Mathematica  (April 15, 2007)