The Matrix Equation Ax = b

The Matrix Equation A\mathbf{x} = \mathbf{b}

The matrix equation A\mathbf{x} = \mathbf{b} is a compact representation of a system of linear equations. In linear algebra, this form is essential for understanding and solving systems using matrix operations.

General Form

The matrix equation is written as:

    \[A\mathbf{x} = \mathbf{b},\]

where:

  • A is an m \times n coefficient matrix,
  • \mathbf{x} is an n \times 1 column vector of variables,
  • \mathbf{b} is an m \times 1 column vector of constants.

This equation corresponds to a system of m linear equations in n variables:

    \[\begin{aligned}     a_{11}x_1 + a_{12}x_2 + \dots + a_{1n}x_n &= b_1, \\     a_{21}x_1 + a_{22}x_2 + \dots + a_{2n}x_n &= b_2, \\     &\vdots \\     a_{m1}x_1 + a_{m2}x_2 + \dots + a_{mn}x_n &= b_m.     \end{aligned}\]

Example

Consider the system:

    \[\begin{aligned}     2x + y &= 5, \\     4x - 3y &= 6.     \end{aligned}\]

In matrix form, this becomes:

    \[\begin{bmatrix} 2 & 1 \\ 4 & -3 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ 6 \end{bmatrix}.\]

Solving A\mathbf{x} = \mathbf{b}

There are several methods to solve the matrix equation:

  • Row Reduction: Use Gaussian elimination to row reduce the augmented matrix:

        \[\begin{bmatrix} A & \mathbf{b} \end{bmatrix}.\]

  • Matrix Inversion: If A is square and invertible, the solution is:

        \[\mathbf{x} = A^{-1}\mathbf{b}.\]

  • LU Decomposition: Factor A into LU, where L is a lower triangular matrix and U is an upper triangular matrix, then solve:

        \[L\mathbf{y} = \mathbf{b}, \quad U\mathbf{x} = \mathbf{y}.\]

Existence and Uniqueness

The solution to A\mathbf{x} = \mathbf{b} depends on the properties of A:

  • Unique Solution: If A is square (m = n) and \det(A) \neq 0, there is a unique solution.
  • No Solution: If \mathbf{b} is not in the column space of A, the system is inconsistent.
  • Infinite Solutions: If A has more columns than rows (n > m), the system may have infinitely many solutions (underdetermined).

Geometric Interpretation

Geometrically, A\mathbf{x} = \mathbf{b} represents a linear transformation. The matrix A transforms the vector \mathbf{x} into the vector \mathbf{b}. The solution determines how to scale and combine the columns of A (its column space) to reach \mathbf{b}.

Applications

The matrix equation A\mathbf{x} = \mathbf{b} is fundamental in numerous fields:

  • Physics: Solving equilibrium problems and systems of forces.
  • Computer Science: Analyzing networks and performing graphics transformations.
  • Engineering: Modeling electrical circuits and mechanical systems.

Summary

The matrix equation A\mathbf{x} = \mathbf{b} is a compact and powerful tool in linear algebra. By understanding its properties and methods of solution, we gain the ability to solve a wide variety of practical and theoretical problems.

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