How a simple matrix transformation reveals the hidden structure of linear systems
In short: Gilbert Strang explains how decomposing a matrix into the product of two smaller matrices, C and R, exposes the fundamental properties of a linear system. This approach simplifies complex data by isolating independent components from dependent ones.
When a matrix A possesses a rank of r, performing elimination results in a row echelon form featuring an identity matrix within its initial r independent columns. The remaining n minus r columns form a matrix F, which acts as a multiplier to derive the dependent columns from the independent ones.
This matrix F is essential because it identifies the basis for both the space spanned by the rows and the set of vectors that map to zero. Ultimately, this process facilitates the factorization of the original matrix into C and R, providing a clearer view of the underlying linear relationships.
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Sources
- Summary by Intellect · source: Elimination and Factorization A = CR
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