Matrices

Compact, elegant, and powerful—matrices organize systems of equations into structured arrays. Essential for linear transformations, computation, and data representation. 🧮

Matrix Notation

Suppose we have the following system of linear equations:

\[ \begin{aligned} 2x_1 - x_2 + x_3 &= 8 \\ x_1 &= 2 \\ 4x_3 &= 7 \end{aligned} \]

We can compactly record the essential information of a linear system in a array called a coefficient matrix:

\[ \begin{bmatrix} 2 & -1 & 1 \\ 1 & 0 & 0 \\ 0 & 0 & 4 \end{bmatrix} \]

This matrix is a squared matrix with dimensions \(n\) x \(n\). Since the matrix has the same number rows (\(3\)) as it has columns (\(3\)).

If constants were known, we can also represent them in a augmented matrix:

\[ \begin{bmatrix} 2 & -1 & 1 & 8\\ 1 & 0 & 0 & 2\\ 0 & 0 & 4 & 7 \end{bmatrix} \]

This matrix is a rectangular matrix with dimensions \(m\) x \(n\). Since the array has the different number rows (\(3\)) and columns (\(4\)).

In data science, it’s common for the constants (outputs) to be unknown or variable. As a result, when we refer to matrices, we’re usually talking about the coefficient matrix.

A more general notation for a matrix denoted \(A\) is

\[ A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} \]

viewof a11 = Inputs.range([-10, 10], {
  step: 1, 
  value: 2, 
  label: tex`a_{11}`, 
  width: 200
})

viewof a12 = Inputs.range([-10, 10], {
  step: 1, 
  value: 3, 
  label: tex`a_{12}`, 
  width: 200
})

viewof a21 = Inputs.range([-10, 10], {
  step: 1, 
  value: 1, 
  label: tex`a_{21}`, 
  width: 200
})

viewof a22 = Inputs.range([-10, 10], {
  step: 1, 
  value: -2, 
  label: tex`a_{22}`, 
  width: 200
})

// Display matrix equation Ax = 0
html`<div style="text-align: center; font-size: 1.3em; margin: 2em 0;">
  <p style="margin-top: 1em;">
    ${tex`A = \begin{bmatrix} ${a11} & ${a12} \\ ${a21} & ${a22} \end{bmatrix}`}
  </p>
</div>`

matrixSolution = {
  // Calculate determinant
  const det = a11 * a22 - a12 * a21;
  
  if (Math.abs(det) < 1e-10) {
    return { status: "Infinite solutions (non-trivial)", x1: null, x2: null, det, inverse: null };
  } else {
    // For homogeneous systems with non-zero determinant, only trivial solution exists
    return { status: "Unique solution (trivial)", x1: 0, x2: 0, det, inverse: null };
  }
}

// Generate points for first line (a11*x + a12*y = 0)
points1 = {
  const result = [];
  for (let i = 0; i < 200; i++) {
    const x = i / 10 - 10;
    if (a12 !== 0) {
      const y = -(a11 * x) / a12;
      if (y >= -10 && y <= 10) {
        result.push({x, y});
      }
    }
  }
  return result;
}

// Generate points for second line (a21*x + a22*y = 0)
points2 = {
  const result = [];
  for (let i = 0; i < 200; i++) {
    const x = i / 10 - 10;
    if (a22 !== 0) {
      const y = -(a21 * x) / a22;
      if (y >= -10 && y <= 10) {
        result.push({x, y});
      }
    }
  }
  return result;
}

// Create the plot
Plot.plot({
  style: "overflow: visible; display: block; margin: 0 auto;",
  width: 500,
  height: 400,
  grid: true,
  x: {label: "x₁", domain: [-10, 10]},
  y: {label: "x₂", domain: [-10, 10]},
  marks: [
    Plot.ruleY([0]),
    Plot.ruleX([0]),
    Plot.line(points1, {
      x: "x", 
      y: "y", 
      stroke: "steelblue",
      strokeWidth: 2
    }),
    Plot.line(points2, {
      x: "x", 
      y: "y", 
      stroke: "crimson",
      strokeWidth: 2
    }),
    matrixSolution.x1 !== null ? Plot.dot([{x: matrixSolution.x1, y: matrixSolution.x2}], {
      x: "x",
      y: "y",
      r: 6,
      fill: "black",
      stroke: "white",
      strokeWidth: 2
    }) : null
  ]
})