SVD: The solution of Ax=b (updated)

 

The solution of Ax=b

A is matrix whose size is m-by-n, b is a n-by-1 column vector. The solution x is a n-by-1 column vector

 

Mldivide is a matlab function, which is also denoted as"\". It solve systems of linear equations Ax = B for x. The method is called left divide.

, The A^-1 is on the left of b, this is the reason called left divide. But the A^-1 exists only when A is a square matrix. When A is not a square matrix, we must apply SVD method to solve this equation. This is how "\" works in matlab.

[m,n] = size(U); U is a column orthogonal matrix

[n,n] = size(s); S is a diagonal matrix

[n,n] = size(V); V is an orthogonal matrix

 

U is a column orthogonal matrix, which means the multiplication of the same column is 1, while the multiplication of different columns is, just likes the following equation:

E is the unit matrix, whose size is n-by-n.

So, we can deduct:

S is a diagonal matrix, so S^-1 is diagonal as well. Each diagonal element of S^-1 is the multiplicative inverse number of the corresponding diagonal element of S.

E is the unit matrix, whose size is n-by-n.

So, we can continue deduct:

The size of U^Tb is n-by-1. We have to stick to the right order of computing.

V is an orthogonal matrix, so

We can get:

VS^-1 can be calculated first, since both of them are square matrixs.

The solution of is .

 

When A is matrix whose size is m-by-n, B is a n-by-k matrix. The solution X is a n-by-k matrix

Solve k equations : using the method above. Each column xⁱ consist the solution X.

 

The matlab code is:

[U,S,V]=svd(A); % Singular valure compsition
x = V/S*(U'*b);

 The above code equals to: 

x=A/b