Abstract: Matrix Completion Problems

Hi! My name is Haoge Chang. I am currently a junior in the college and I am majoring in Applied Mathematics and Economics. I will be conducting research on matrix completion problems this summer, focusing on the completion problems of totally nonnegative matrices and totally positive matrices.  I hope that through my research I could discover more interesting properties of these problems.

To start my blog, I will briefly introduce the definition of matrices, determinants, submatrices and minors.

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Matrices are widely used in other branches of mathematics, natural sciences and social sciences.  They are mostly used to represent linear equation systems. For example, a linear system of 3×1+2×2=5 and 4×1+5×2=10 can be represented as:

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A matrix can also be a place to hold information. For example, a dataset with m observation and n categories can be represented as a m*n matrix (a matrix with m rows and n columns). Another example is an 2×2 variance-covariance matrix.

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Now I will introduce the definition of a determinant. The determinant is a useful value that can be calculated from the entries of a matrix. This special value has many applications, ranging from calculating the area in a 2D plane to checking if a group of data is explosive or not. For our matrix above, the determinant is calculated as:

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Now I am going to introduce submatrices and minors. To do this, I need a larger matrix:

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Suppose I randomly delete one row and one column of this matrix. I will be left with a 2×2 matrix. This smaller matrix is called a submatrix of the original matrix. The determinant of each submatrix is called a minor. Because there are 9 ways to choose which row and column to delete, we have nine 2×2 submatrices and nine subsequent minors associated with each submatrix (Notice we also have 9 1×1 submatrices). A totally positive (or nonnegative) matrix is a matrix such that all its minors (1×1, 2×2, 3×3…) are positive (or nonnegative).

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In my research, I will be working on matrix completion problems, focusing on totally positive and totally nonnegative matrices: given a matrix with some unspecified entries, we are interested in whether this matrix will become a totally nonnegative matrix or totally positive matrix, depending on the values chosen for such unspecified entries. For example, if we are handed a matrix that looks like:

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? represents the place we have a unspecified entry. We can ask many questions regarding this entry. Can we find a value for the unspecified entry such that all minors of this matrix are positive (or nonnegative)? What are the general properties the specified entries must have to accommodate a value for the unspecified entry? This question will become even more intriguing if one changes the size (inserting more rows and columns, making it irregular) and changes the positions and the patterns of the unspecified entry. In my research, I will try to unveil interesting properties that are associated with these problems.

Thank you for reading my blog. You can follow my progress through this blog. I am really looking forward to conducting this research and finding interesting results this summer!