Non-Square Matrices
A non-square matrix is a matrix where the number of rows is not equal to the number of columns. These matrices are often described as “tall” or “fat” depending on their dimensions.
- A tall matrix has more rows than columns (e.g., a \(3 \times 2\) matrix).
- A fat matrix has more columns than rows (e.g., a \(2 \times 3\) matrix).
The shape of a matrix is particularly important when analyzing systems of linear equations.
Non-Square Matrices in Linear Systems
Consider a system of linear equations represented as:
\[Ax = b\]Here, \(A\) is the coefficient matrix, \(x\) is the vector of unknowns, and \(b\) is the resulting vector. The dimensions of \(A\) determine the nature of the solution.
Overdetermined Systems (Tall Matrix)
An overdetermined system has more equations than unknowns. This occurs when the matrix \(A\) is tall.
- Shape: \(A\) is \(m \times n\), \(x\) is \(n \times 1\), and \(b\) is \(m \times 1\), with \(m > n\).
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Example: Let \(A\) be a \(3 \times 2\) matrix. This system has 3 equations and only 2 unknown variables.
\[A_{3 \times 2} \cdot x_{2 \times 1} = b_{3 \times 1}\] - Implication: Because there are more constraints (equations) than variables, it is unlikely that a single solution for \(x\) can satisfy all equations simultaneously. Such a system typically has no exact solution.
Underdetermined Systems (Fat Matrix)
An underdetermined system has fewer equations than unknowns. This occurs when the matrix \(A\) is fat.
- Shape: \(A\) is \(m \times n\), \(x\) is \(n \times 1\), and \(b\) is \(m \times 1\), with \(m < n\).
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Example: Let \(A\) be a \(2 \times 3\) matrix. This system has 2 equations but 3 unknown variables.
\[A_{2 \times 3} \cdot x_{3 \times 1} = b_{2 \times 1}\] - Implication: With more variables than constraints, there isn’t enough information to pinpoint a single solution. If a solution exists, there are typically infinitely many solutions.
Summary
The relationship between the matrix shape and the nature of the solution for \(Ax=b\) can be summarized as follows:
| System Type | Matrix Shape (\(A_{m \times n}\)) | Description | Typical Solution |
|---|---|---|---|
| Overdetermined | Tall (\(m > n\)) | More equations than unknowns | No exact solution (or a unique solution in rare cases) |
| Underdetermined | Fat (\(m < n\)) | Fewer equations than unknowns | Infinitely many solutions (or no solution) |
| Determined (Square) | Square (\(m = n\)) | Equal equations and unknowns | A unique solution (if \(A\) is invertible) |