matrix_decomposition
Decompose matrices using eigenvalues/vectors, SVD, QR, Cholesky, or LU factorisation for linear algebra analysis and computational workflows.
Instructions
Matrix decompositions: eigenvalues/vectors, SVD, QR, Cholesky, LU.
Examples:
EIGENVALUE DECOMPOSITION: matrix=[[4,2],[1,3]], decomposition="eigen" Result: {eigenvalues: [5, 2], eigenvectors: [[0.89,0.45],[0.71,-0.71]]}
SINGULAR VALUE DECOMPOSITION (SVD): matrix=[[1,2],[3,4],[5,6]], decomposition="svd" Result: {U: 3×3, singular_values: [9.5, 0.77], Vt: 2×2}
QR FACTORISATION: matrix=[[1,2],[3,4]], decomposition="qr" Result: {Q: orthogonal, R: upper triangular}
CHOLESKY (symmetric positive definite): matrix=[[4,2],[2,3]], decomposition="cholesky" Result: {L: [[2,0],[1,1.41]]} where A=LL^T
LU DECOMPOSITION: matrix=[[2,1],[4,3]], decomposition="lu" Result: {P: permutation, L: lower, U: upper} where A=PLU
Input Schema
| Name | Required | Description | Default |
|---|---|---|---|
| context | No | Optional annotation to label this calculation (e.g., 'Bond A PV', 'Q2 revenue'). Appears in results for easy identification. | |
| output_mode | No | Output format: full (default), compact, minimal, value, or final. See batch_execute tool for details. | full |
| matrix | Yes | Matrix to decompose as 2D nested list (e.g., [[4,2],[1,3]]) | |
| decomposition | Yes | Decomposition type: eigen=eigenvalues/vectors, svd=singular value, qr=QR, cholesky=symmetric positive definite, lu=LU factorisation |
Input Schema (JSON Schema)
{
"properties": {
"context": {
"anyOf": [
{
"type": "string"
},
{
"type": "null"
}
],
"default": null,
"description": "Optional annotation to label this calculation (e.g., 'Bond A PV', 'Q2 revenue'). Appears in results for easy identification."
},
"decomposition": {
"description": "Decomposition type: eigen=eigenvalues/vectors, svd=singular value, qr=QR, cholesky=symmetric positive definite, lu=LU factorisation",
"enum": [
"eigen",
"svd",
"qr",
"cholesky",
"lu"
],
"type": "string"
},
"matrix": {
"description": "Matrix to decompose as 2D nested list (e.g., [[4,2],[1,3]])",
"items": {
"items": {
"type": "number"
},
"type": "array"
},
"type": "array"
},
"output_mode": {
"default": "full",
"description": "Output format: full (default), compact, minimal, value, or final. See batch_execute tool for details.",
"enum": [
"full",
"compact",
"minimal",
"value",
"final"
],
"type": "string"
}
},
"required": [
"matrix",
"decomposition"
],
"type": "object"
}
Implementation Reference
- Handler function that performs various matrix decompositions (eigen, SVD, QR, Cholesky, LU) based on the specified decomposition type using SciPy's linalg module.async def matrix_decomposition( matrix: Annotated[List[List[float]], Field(description="Matrix to decompose as 2D nested list (e.g., [[4,2],[1,3]])")], decomposition: Annotated[Literal["eigen", "svd", "qr", "cholesky", "lu"], Field(description="Decomposition type: eigen=eigenvalues/vectors, svd=singular value, qr=QR, cholesky=symmetric positive definite, lu=LU factorisation")], ) -> str: """Matrix decompositions using SciPy: eigen (λ,v), SVD (UΣV^T), QR (orthogonal×triangular), Cholesky (LL^T), LU (PLU). For analysis, solving, and numerical stability.""" try: mat = list_to_numpy(matrix) if decomposition == "eigen": if mat.shape[0] != mat.shape[1]: raise ValueError( f"Eigenvalue decomposition requires square matrix. Got shape: {mat.shape}" ) eigenvalues: NDArray[np.complexfloating] eigenvectors: NDArray[np.complexfloating] eigenvalues, eigenvectors = la.eig(mat) # type: ignore[misc] return format_result( { "eigenvalues": eigenvalues.tolist(), "eigenvectors": eigenvectors.tolist(), }, {"decomposition": decomposition} ) elif decomposition == "svd": U, s, Vt = la.svd(mat) return format_result( { "U": U.tolist(), "singular_values": s.tolist(), "Vt": Vt.tolist(), }, {"decomposition": decomposition} ) elif decomposition == "qr": Q: NDArray[np.floating] R: NDArray[np.floating] Q, R = la.qr(mat) # type: ignore[misc] return format_result( {"Q": Q.tolist(), "R": R.tolist()}, {"decomposition": decomposition} ) elif decomposition == "cholesky": if mat.shape[0] != mat.shape[1]: raise ValueError( f"Cholesky decomposition requires square matrix. Got shape: {mat.shape}" ) # Check if matrix is symmetric if not np.allclose(mat, mat.T): raise ValueError("Cholesky decomposition requires symmetric matrix") try: L = la.cholesky(mat, lower=True) return format_result( {"L": L.tolist()}, {"decomposition": decomposition, "note": "A = L * L^T"} ) except np.linalg.LinAlgError: raise ValueError("Matrix is not positive definite") elif decomposition == "lu": P, L, U = la.lu(mat) # type: ignore[misc] return format_result( { "P": P.tolist(), "L": L.tolist(), "U": U.tolist(), }, {"decomposition": decomposition, "note": "A = P * L * U"} ) else: raise ValueError(f"Unknown decomposition: {decomposition}") except Exception as e: if isinstance(e, ValueError): raise raise ValueError(f"Matrix decomposition failed: {str(e)}")
- src/vibe_math_mcp/tools/linalg.py:181-211 (registration)MCP tool registration for 'matrix_decomposition' including name, description, examples, and annotations.@mcp.tool( name="matrix_decomposition", description="""Matrix decompositions: eigenvalues/vectors, SVD, QR, Cholesky, LU. Examples: EIGENVALUE DECOMPOSITION: matrix=[[4,2],[1,3]], decomposition="eigen" Result: {eigenvalues: [5, 2], eigenvectors: [[0.89,0.45],[0.71,-0.71]]} SINGULAR VALUE DECOMPOSITION (SVD): matrix=[[1,2],[3,4],[5,6]], decomposition="svd" Result: {U: 3×3, singular_values: [9.5, 0.77], Vt: 2×2} QR FACTORISATION: matrix=[[1,2],[3,4]], decomposition="qr" Result: {Q: orthogonal, R: upper triangular} CHOLESKY (symmetric positive definite): matrix=[[4,2],[2,3]], decomposition="cholesky" Result: {L: [[2,0],[1,1.41]]} where A=LL^T LU DECOMPOSITION: matrix=[[2,1],[4,3]], decomposition="lu" Result: {P: permutation, L: lower, U: upper} where A=PLU""", annotations=ToolAnnotations( title="Matrix Decomposition", readOnlyHint=True, idempotentHint=True, ), )
- Input schema definitions using Pydantic Annotated types for matrix (List[List[float]]) and decomposition type (Literal).matrix: Annotated[List[List[float]], Field(description="Matrix to decompose as 2D nested list (e.g., [[4,2],[1,3]])")], decomposition: Annotated[Literal["eigen", "svd", "qr", "cholesky", "lu"], Field(description="Decomposition type: eigen=eigenvalues/vectors, svd=singular value, qr=QR, cholesky=symmetric positive definite, lu=LU factorisation")], ) -> str: