Column Echelon Form Calculator

Column Echelon Form Calculator

The Column Echelon Form (CEF) of a matrix is a specific arrangement that simplifies the matrix while preserving key properties related to its column space. It's analogous to the Row Echelon Form (REF) but applied to columns instead of rows. This calculator will help you transform any 3×3 matrix into its Column Echelon Form.

Input Matrix (3×3)

Enter the elements of your 3×3 matrix below. You can use integers or decimal numbers. Empty fields will be treated as 0.

Resulting Column Echelon Form

Understanding Column Echelon Form

Column Echelon Form (CEF) is a canonical form for a matrix, similar to Row Echelon Form (REF) but with the roles of rows and columns swapped. A matrix is in Column Echelon Form if it satisfies the following conditions:

  1. All zero columns are to the right of all non-zero columns.
  2. For each non-zero column, the first non-zero entry (from the top) is called the leading entry or pivot.
  3. For any two non-zero columns, the leading entry of the column to the left is in a row above the leading entry of the column to the right. This creates a "staircase" pattern when viewed column-wise.
  4. All entries below a leading entry in its column are zero.

Why is Column Echelon Form Important?

Column Echelon Form is a powerful tool in linear algebra with several applications:

  • Column Space: It helps in finding a basis for the column space of a matrix. The columns of the original matrix that correspond to the pivot columns in its Column Echelon Form form a basis for the column space.
  • Linear Independence: It can be used to determine if a set of vectors (when arranged as columns of a matrix) is linearly independent.
  • Matrix Properties: Understanding the structure of a matrix in CEF can reveal insights into its rank and other fundamental properties.

How to Calculate Column Echelon Form

The most common and straightforward method to find the Column Echelon Form of a matrix involves three steps:

  1. Transpose the Original Matrix: Convert the given matrix A into its transpose, AT. This means rows become columns and columns become rows.
  2. Apply Row Echelon Form (REF) to the Transposed Matrix: Use Gaussian elimination (row operations) to transform AT into its Row Echelon Form, REF(AT). Row operations include swapping rows, multiplying a row by a non-zero scalar, and adding a multiple of one row to another.
  3. Transpose Back: Finally, transpose the resulting REF(AT) matrix back. This final matrix is the Column Echelon Form of the original matrix A.

Using This Calculator

To use the calculator, simply input the numerical values for each element of your 3×3 matrix into the provided fields. Click the "Calculate Column Echelon Form" button to see the result. The "Clear" button will reset all input fields and the output.

Example

Let's consider the following matrix:

1 2 3
4 5 6
7 8 9

When you input these values into the calculator and click "Calculate", the steps internally would be:

  1. Transpose:
    147
    258
    369
  2. Row Echelon Form of Transposed Matrix:
    147
    012
    000
  3. Transpose Back (Column Echelon Form):
    100
    410
    720

This calculator provides a quick and accurate way to find the Column Echelon Form for your matrices, aiding in your linear algebra studies and applications.

function transposeMatrix(m) { var numRows = m.length; if (numRows === 0) return []; var numCols = m[0].length; if (numCols === 0) return m; var result = []; for (var j = 0; j < numCols; j++) { var newRow = []; for (var i = 0; i < numRows; i++) { newRow.push(m[i][j]); } result.push(newRow); } return result; } function rowEchelonForm(matrix) { var numRows = matrix.length; if (numRows === 0) return []; var numCols = matrix[0].length; if (numCols === 0) return matrix; var r = 0; // current row var c = 0; // current column (pivot column) // Create a deep copy to avoid modifying the original matrix reference var currentMatrix = matrix.map(function(arr) { return arr.slice(); }); while (r < numRows && c < numCols) { // Find pivot row: find a row 'i' such that currentMatrix[i][c] is non-zero, starting from current row 'r' var pivotRow = r; for (var i = r + 1; i Math.abs(currentMatrix[pivotRow][c])) { pivotRow = i; } } // If the pivot element is zero (or very close to zero), move to the next column if (Math.abs(currentMatrix[pivotRow][c]) < 1e-9) { // Use a small epsilon for floating point comparison c++; continue; } // Swap current row 'r' with pivotRow var tempRow = currentMatrix[r]; currentMatrix[r] = currentMatrix[pivotRow]; currentMatrix[pivotRow] = tempRow; // Make the pivot element 1 (divide row 'r' by pivot) var pivot = currentMatrix[r][c]; for (var j = c; j < numCols; j++) { currentMatrix[r][j] /= pivot; } // Eliminate other entries below the pivot in the current column 'c' for (var i = r + 1; i < numRows; i++) { // Only below the pivot for REF var factor = currentMatrix[i][c]; for (var j = c; j < numCols; j++) { currentMatrix[i][j] -= factor * currentMatrix[r][j]; } } r++; // Move to the next row c++; // Move to the next column } return currentMatrix; } function displayMatrix(matrix, elementId) { var outputDiv = document.getElementById(elementId); if (!matrix || matrix.length === 0) { outputDiv.innerHTML = "No matrix to display."; return; } var html = ''; for (var i = 0; i < matrix.length; i++) { html += ''; for (var j = 0; j < matrix[i].length; j++) { // Format numbers to a reasonable precision, treating very small numbers as 0 var val = matrix[i][j]; var formattedVal = Math.abs(val) < 1e-9 ? 0 : val.toFixed(4); html += ''; } html += ''; } html += '
' + formattedVal + '
'; outputDiv.innerHTML = html; } function calculateColumnEchelonForm() { var numRows = 3; var numCols = 3; var matrix = []; var errorMessageDiv = document.getElementById("errorMessage"); errorMessageDiv.innerHTML = ""; // Clear previous errors // Read input matrix for (var i = 0; i < numRows; i++) { var row = []; for (var j = 0; j < numCols; j++) { var inputId = "m" + i + j; var value = document.getElementById(inputId).value; var num = parseFloat(value); if (value.trim() === "") { // Allow empty for 0 row.push(0); } else if (isNaN(num)) { errorMessageDiv.innerHTML = "Error: Please enter valid numbers for all matrix elements."; document.getElementById("resultMatrix").innerHTML = ""; return; } else { row.push(num); } } matrix.push(row); } // Deep copy for calculations var originalMatrixCopy = matrix.map(function(arr) { return arr.slice(); }); // Step 1: Transpose the original matrix var transposed = transposeMatrix(originalMatrixCopy); // Step 2: Apply Row Echelon Form to the transposed matrix var refTransposed = rowEchelonForm(transposed); // Step 3: Transpose back to get Column Echelon Form var columnEchelon = transposeMatrix(refTransposed); // Display result displayMatrix(columnEchelon, "resultMatrix"); } function clearMatrixInputs() { var ids = ["m00", "m01", "m02", "m10", "m11", "m12", "m20", "m21", "m22"]; for (var i = 0; i < ids.length; i++) { document.getElementById(ids[i]).value = ""; } document.getElementById("resultMatrix").innerHTML = ""; document.getElementById("errorMessage").innerHTML = ""; }

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