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Deepbox
20intermediate

Linear Algebra Operations

Determinant, inverse, norms, matrix power, SVD, QR, LU, symmetric eigendecomposition and linear systems. Each decomposition is checked by multiplying the factors back together.

What you will learn

  • Use deepbox/linalg for det, inv, trace, norm, matrixPower, svd, qr, lu, eigh, solve.
  • Use deepbox/ndarray for tensor, diag, matmul, dot, .T.
  • Determinant, inverse, norms, matrix power, SVD, QR, LU, symmetric eigendecomposition and linear systems. Each decomposition is checked by multiplying the factors back together.

Source

index.ts
/** * Example 20: Linear Algebra Operations * * Determinant, inverse, norms, matrix decompositions (SVD, QR, LU, eigen) and * linear systems. Each decomposition is checked by multiplying the factors * back together and comparing with the original matrix. */ import { det, eigh, inv, lu, matrixPower, norm, qr, solve, svd, trace } from "deepbox/linalg";import { diag, type Tensor, tensor } from "deepbox/ndarray"; // Largest absolute difference between two tensors of the same shape.const maxError = (a: Tensor, b: Tensor): string =>  Number(a.sub(b).abs().max().item()).toExponential(2); console.log("=== Linear Algebra Operations ===\n"); // Create a matrixconst A = tensor([  [4, 2],  [3, 1],]); console.log("Matrix A:");console.log(`${A.toString()}\n`); // Determinant (a plain number)console.log(`Determinant: ${det(A).toFixed(4)}\n`); // Trace (a tensor, read with item())console.log(`Trace: ${Number(trace(A).item()).toFixed(4)}\n`); // Matrix inverse, checked by multiplying with Aconst invA = inv(A);console.log("Inverse of A:");console.log(`${invA.toString()}\n`);console.log(`Largest error in A * inv(A) - I: ${maxError(A.matmul(invA), diag(tensor([1, 1])))}\n`); // Matrix norms (plain numbers)console.log(`Frobenius norm: ${norm(A, "fro").toFixed(4)}\n`); // Integer matrix power: A^3 = A * A * Aconsole.log("A cubed with matrixPower(A, 3):");console.log(`${matrixPower(A, 3).toString()}\n`); // SVD Decompositionconsole.log("SVD Decomposition:");console.log("-".repeat(50)); const B = tensor([  [1, 2],  [3, 4],  [5, 6],]); // The second argument false asks for the reduced SVD: U is [3, 2] instead of [3, 3].const [U, S, Vt] = svd(B, false);console.log("U (left singular vectors):");console.log(U.toString());console.log("\nS (singular values):");console.log(S.toString());console.log("\nVt (right singular vectors transposed):");console.log(`${Vt.toString()}\n`);console.log(  `Largest error in U * diag(S) * Vt - B: ${maxError(U.matmul(diag(S)).matmul(Vt), B)}\n`); // QR Decompositionconsole.log("QR Decomposition:");console.log("-".repeat(50)); const [Q, R] = qr(B);console.log("Q (orthonormal columns):");console.log(Q.toString());console.log("\nR (upper triangular):");console.log(`${R.toString()}\n`);console.log(`Largest error in Q * R - B: ${maxError(Q.matmul(R), B)}\n`); // LU Decompositionconsole.log("LU Decomposition:");console.log("-".repeat(50)); const D = tensor([  [1, 2],  [3, 4],]); // P is a permutation matrix and D = P * L * U.const [P, L, Ulu] = lu(D);console.log("P (permutation):");console.log(P.toString());console.log("\nL (lower triangular):");console.log(L.toString());console.log("\nU (upper triangular):");console.log(Ulu.toString());console.log(`\nLargest error in P * L * U - D: ${maxError(P.matmul(L).matmul(Ulu), D)}\n`); // Eigendecomposition of a symmetric matrixconsole.log("Eigendecomposition (symmetric):");console.log("-".repeat(50)); const M = tensor([  [2, 1],  [1, 2],]);// Eigenvalues come back in ascending order. Eigenvectors are the columns of V.const [eigenvalues, V] = eigh(M);console.log("Eigenvalues:", eigenvalues.toString());console.log("Eigenvectors (columns):");console.log(V.toString());console.log(  `Largest error in V * diag(w) * V^T - M: ${maxError(V.matmul(diag(eigenvalues)).matmul(V.T), M)}\n`); // Solving linear systems: Ax = bconsole.log("Solving Linear System Ax = b:");console.log("-".repeat(50)); const ASys = tensor([  [3, 1],  [1, 2],]);const b = tensor([9, 8]); // dot() multiplies a matrix by a vector. matmul() needs two 2D tensors.const x = solve(ASys, b);console.log("Solution x:");console.log(x.toString());console.log(`Largest error in A * x - b: ${maxError(ASys.dot(x), b)}`);

Output

Console output
Console output only.