- Let $A$ be a symmetric matrix belonging to $M_n(\mathbb{R})$. If the rank of $A$ is denoted by $r$, demonstrate the existence of indices $i_1, i_2,\dots,i_r$ such that the $r\times r$ sub-matrix, formed by the $i_1,i_2,\dots,i_r$ rows and $i_1,i_2,\dots,i_r$ columns of matrix $A$, is invertible.
- Further, illustrate that $A$ can be expressed as the sum of $r$ matrices, each having a rank of 1.
For the initial query, it seems intuitive to select $r$ linearly independent rows, thereby rendering the sub-matrix invertible. However, the second part of the question presents a challenge. My knowledge from a course in numerical analysis suggests the potential utility of Singular Value Decomposition (SVD), but its application is restricted in this context as the problem originates from a Linear Algebra course. My search for solutions primarily yielded answers involving SVD. Based on the textbook by Hoffman and Kunze, I am allowed to utilize the following concepts:
- Linear Equations (including key aspects such as elementary row operations, RREF form)
- Vector Spaces
- Linear Transformations (including key aspects such as representation of transformations by matrices, linear functionals, the double dual)
I hypothesize that the most appropriate approach would involve the consideration that the dimensions of the row space and column space of a matrix are equal. Despite attempts to use the linearly independent rows and columns of $A$, I have yet to arrive at a useful conclusion.
In addition, I would appreciate if someone could provide a clear explanation of the characteristics of rank 1 matrices and propose a basic method to tackle this problem.
Any assistance or hints would be greatly appreciated!