The proof here is about the consistency of the system of normal equations $X^T X B = X^T Y$.

Given:

1. Definitions: $A = X^T X$ and $a = X^T Y$.
2. Objective: Show that the rank $r(X^T X) = r([X^T X, X^T Y])$, which implies the system is consistent.

Step-by-Step Explanation:

1. Initial Setup: The proof starts with the notation $A = X^T X$ and $a = X^T Y$, where we need to show that the system of normal equations $X^T X B = X^T Y$ is consistent. This requires showing that the rank of $X^T X$ remains the same even when the column $X^T Y$ is added.

2. Rank Inequality: It establishes that $r(X^T X) \leq r([X^T X, X^T Y])$. This is true because adding another column (like $X^T Y$) can either keep the rank the same or increase it, but it will not decrease.

3. Further Rank Constraints:

   • The next inequality is $r([X^T X, X^T Y]) \leq \min(r(X^T), r(X^T Y))$.
   • This implies that the rank of the matrix formed by $X^T X$ and $X^T Y$ is constrained by the smaller of the two ranks: $r(X^T)$ and $r(X^T Y)$.

4. Final Comparison: The final sequence of inequalities concludes with $r(X) = r(X^T X)$. Since $r(X^T X) = r([X^T X, X^T Y])$, it implies that the ranks are equal, ensuring the system is consistent.

5. Conclusion:

   • Since $r(X^T X) = r([X^T X, X^T Y]) = r$, the normal equations are consistent, which means solutions always exist. However, uniqueness is not guaranteed; there may be multiple solutions depending on the rank of $X$.

In summary, the proof demonstrates that the system of normal equations $X^T X B = X^T Y$ is consistent because the addition of $X^T Y$ does not change the rank of $X^T X$.

Would you like a deeper breakdown of each inequality used, or have any further questions?

Related Questions

1. What does it mean for a system of linear equations to be consistent?
2. How is the rank of a matrix defined, and why is it important in linear algebra?
3. Why do normal equations $X^T X B = X^T Y$ arise in least squares problems?
4. How does one interpret the rank of a matrix in terms of the number of solutions to a linear system?
5. What is the difference between a consistent system and a unique solution?

Tip

For a linear system to be consistent, the rank of the coefficient matrix should match the rank of the augmented matrix. This ensures at least one solution exists.