موجه

make a video depicting a math professor explaining to his student the following concepts: The **rank function**, denoted as $A \mapsto \text{rank}(A)$, is defined as the function that associates any matrix $A$ with its rank. The rank of a matrix is equivalent to the number of its **non-zero singular values** or the number of its **linearly independent rows or columns**. While this function is integer-valued and possesses the property of being **lower semi-continuous**, it is **not continuous**. ### Evidence of Non-Continuity The rank function's lack of continuity is primarily due to its sensitivity to small changes in matrix elements: * **Limit of Sequences:** A sequence of matrices can have a constant rank, yet converge to a matrix with a different rank. For example, in the sequence of matrices $\begin{pmatrix} 1/n & 0 \\ 0 & 1/n \end{pmatrix}$, every individual matrix has a rank of 2, but the limit of the sequence is the zero matrix, which has a rank of 0. * **Ill-Posedness:** The rank is considered **ill-posed** because it does not depend on matrix elements in a Lipschitz continuous manner. A tiny perturbation in a matrix $A$ can change its rank instantly, whereas continuous functions do not exhibit such abrupt jumps from infinitesimal changes. * **Neighborhood Properties:** In any neighborhood of a matrix $A$ with rank $k$, there exist matrices with ranks of $k+1, k+2, \dots, p$. Because the rank can jump to a higher value even in an arbitrarily small neighborhood, the function fails the criteria for continuity. ### Lower Semi-Continuity Although not continuous, the rank function is **lower semi-continuous** everywhere. This means that if a sequence of matrices $A_\nu$ converges to $A$, the rank of the limit is at most the limit of the ranks: $\liminf_{\nu \to \infty} \text{rank}(A_\nu) \geq \text{rank}(A)$. This property is derived from the fact that a matrix has a rank of at least $r$ if and only if it contains an invertible $r \times r$ submatrix. Because the **determinant function is continuous**, an invertible submatrix will remain invertible in a sufficiently small neighborhood of the original matrix, ensuring that the rank **does not decrease** in that neighborhood.

نتيجة

make a video depicting a math professor explaining to his student the following concepts: The **rank function**, deno...

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