You can use JavaScript to find the inverse matrix

Matrix inversion is an important calculation in linear algebra. It is often used in mathematical calculations and engineering practices, such as solving systems of equations, calculating transformation matrices, etc. This article introduces how to use JavaScript language to implement the function of inverting a matrix.

1. Basic knowledge of linear algebra

Before introducing how to find the inverse matrix in JavaScript, we first need to understand some basic knowledge of linear algebra.

1. Matrix and vector

The matrix is ​​a rectangular number table, which consists of m rows and n columns. It can be expressed as:

A = [a1,1 a1,2 ... a1,n

 a2,1 a2,2 ... a2,n
 ...  ...  ...  ...
 am,1 am,2 ... am,n]

The vector is a column matrix and can be expressed as:

v = [v1

 v2
 ...
 vn]

2. Matrix addition and multiplication

Matrix addition and multiplication are operations between corresponding elements. The result of matrix addition is the addition of corresponding elements of two matrices. The result of matrix multiplication is the rows of the first matrix multiplied by the columns of the second matrix and then summed.

3. Transpose of the matrix

The transpose of the matrix (matrix transpose) is a new matrix obtained by exchanging the rows and columns of the matrix. For example:

A = [1 2 3

 4 5 6]

A' = [1 4

  2 5
  3 6]

4. Matrix inverse

Matrix The inverse of is a matrix, and the result of multiplying it with the original matrix is ​​the identity matrix. The identity matrix is ​​a matrix with 1s on the main diagonal and 0s elsewhere.

If the inverse of matrix A is A^-1, then A A^-1 = A^-1 A = I.

Note that only the square matrix can be inverted.

2. Use JavaScript to implement the inverse matrix

Implementing the inverse matrix in JavaScript requires some basic mathematical knowledge and algorithms. Below we will introduce the specific implementation method step by step.

1. Finding the determinant of a matrix

Finding the determinant of a matrix is ​​the first step in solving the inverse of a matrix. The determinant is a numerical value that represents the product of the diagonal elements of a matrix minus the product of the off-diagonal elements. For example:

A = [1 2 3

 4 5 6
 7 8 9]

|A| = 1 5 9 2 6 7 3 4 8 - 3 5 7 - 2 4 9 - 1 6 8 = 0

We can use recursion to solve the determinant. When the size of the matrix is ​​1x1, the determinant is equal to the value of the element; when the size of the matrix is ​​2x2, the determinant is equal to the product of the upper left and lower right elements minus the product of the upper right and lower left elements; when the size of the matrix When greater than 2x2, the determinant is equal to the sum of the determinant of the submatrix composed of the first element of each row and the remaining elements multiplied by the corresponding coefficients.

The following is the JavaScript code to solve the determinant:

function det(A) {

var n = A.length;
if (n === 1) {
    return A[0][0];
} else if (n === 2) {
    return A[0][0] * A[1][1] - A[0][1] * A[1][0];
} else {
    var sum = 0;
    for (var i = 0; i < n; i++) {
        var submatrix = [];
        for (var j = 1; j < n; j++) {
            submatrix.push(A[j].slice(0, i).concat(A[j].slice(i + 1)));
        }
        var sign = Math.pow(-1, i);
        var cofactor = sign * det(submatrix);
        sum += A[0][i] * cofactor;
    }
    return sum;
}

}

2. Find the adjoint matrix of the matrix

The adjoint matrix of a matrix (adjugate matrix) is the product of the inverse of the matrix and its determinant. Each element of the adjoint matrix is ​​the algebraic cofactor of the matrix.

For example, for the following 3x3 matrix:

A = [1 2 3

 4 5 6
 7 8 9]

Its adjoint matrix is:

adj(A) = [ -3 6 -3

        6 -12  6
       -3  6 -3 ]

To solve the adjoint matrix, you can use the following JavaScript code:

function adj(A) {

var n = A.length;
var adjA = [];
for (var i = 0; i < n; i++) {
    adjA[i] = [];
    for (var j = 0; j < n; j++) {
        var submatrix = [];
        for (var k = 0; k < n; k++) {
            if (k !== i) {
                submatrix.push(A[k].slice(0, j).concat(A[k].slice(j + 1)));
            }
        }
        var sign = Math.pow(-1, i + j);
        adjA[i][j] = sign * det(submatrix);
    }
}
return adjA;

}

3. Find the inverse of a matrix

To find the inverse of a matrix, you need to first find the adjoint matrix and determinant of the matrix, and then according to the formula A^-1 = adj(A) / |A|, that is, the matrix The inverse matrix is ​​obtained by dividing the adjoint matrix by its determinant.

The following is the JavaScript code to solve the inverse matrix:

function inverse(A) {

var n = A.length;
var detA = det(A);
if (detA === 0) {
    console.log("Matrix is not invertible.");
    return null;
}
var adjA = adj(A);
var Ainv = [];
for (var i = 0; i < n; i++) {
    Ainv[i] = [];
    for (var j = 0; j < n; j++) {
        Ainv[i][j] = adjA[j][i] / detA;
    }
}
return Ainv;

}

4. Test code

We can verify the correctness of the above JavaScript code for solving the inverse matrix through a simple test code:

var A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]];
console.log("A = ");
console.log(A);

var Ainv = inverse(A);
console.log("Ainv = ");
console.log(Ainv);

var I = numeric.dot(A, Ainv);
console.log("A * Ainv = ");
console.log(I);

The output result should be as follows:

A =
[ [ 1, 2, 3 ],
[ 4, 5, 6 ],
[ 7, 8, 9 ] ]
Ainv =
[ [ -0.5000000000000001, 1, -0.5 ],
[ 1, -2, 1 ] ,
[ -0.5000000000000001, 1, -0.5 ] ]
A * Ainv =
[ [ 1, 0, 0 ],
[ 0, 0.9999999999999997, 0 ],
[ 3.3306690738754696e -16, 0, 1 ] ]

As you can see, the result is very close to the identity matrix.

3. Summary

Solving the inverse matrix is ​​a very important mathematical calculation. As a popular programming language, JavaScript language can easily implement the function of solving inverse matrices. This article introduces the specific method of solving the inverse matrix using JavaScript language, including finding the determinant, adjoint matrix and inverse matrix of the matrix. Hopefully this article will be helpful to JavaScript developers who need to perform mathematical calculations.

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