Linear Algebra

def dot(a, b, out=None):
    """Dot product of two arrays.
    
    Args:
        a: First input array
        b: Second input array  
        out: Output array
        
    Returns:
        cupy.ndarray: Dot product result
    """

def matmul(x1, x2, out=None, **kwargs):
    """Matrix product of two arrays.
    
    Args:
        x1, x2: Input arrays
        out: Output array
        **kwargs: Additional arguments
        
    Returns:
        cupy.ndarray: Matrix product
    """

def inner(a, b):
    """Inner product of two arrays."""

def outer(a, b, out=None):
    """Compute outer product of two vectors."""

def tensordot(a, b, axes=2):
    """Compute tensor dot product along specified axes."""

def kron(a, b):
    """Kronecker product of two arrays."""

def norm(x, ord=None, axis=None, keepdims=False):
    """Matrix or vector norm.
    
    Args:
        x: Input array
        ord: Order of norm (None, 'fro', 'nuc', inf, -inf, 1, -1, 2, -2)
        axis: Axis for norm calculation
        keepdims: Keep reduced dimensions
        
    Returns:
        cupy.ndarray or float: Norm value(s)
    """

def trace(a, offset=0, axis1=0, axis2=1, dtype=None, out=None):
    """Return sum along diagonals of array.
    
    Args:
        a: Input array
        offset: Diagonal offset
        axis1, axis2: Axes to take diagonal from
        dtype: Output data type
        out: Output array
        
    Returns:
        cupy.ndarray: Sum of diagonal elements
    """

def det(a):
    """Compute determinant of array.
    
    Args:
        a: Input array
        
    Returns:
        cupy.ndarray: Determinant
    """

def slogdet(a):
    """Compute sign and logarithm of determinant.
    
    Args:
        a: Input array
        
    Returns:
        tuple: (sign, logdet) arrays
    """

def matrix_rank(M, tol=None, hermitian=False):
    """Return matrix rank of array using SVD."""

def solve(a, b):
    """Solve linear matrix equation ax = b.
    
    Args:
        a: Coefficient matrix
        b: Ordinate values
        
    Returns:
        cupy.ndarray: Solution to system
    """

def lstsq(a, b, rcond=None):
    """Return least-squares solution to linear matrix equation.
    
    Args:
        a: Coefficient matrix
        b: Ordinate values
        rcond: Cutoff for small singular values
        
    Returns:
        tuple: (solution, residuals, rank, singular_values)
    """

def solve_triangular(a, b, lower=False, unit_diagonal=False, overwrite_b=False, debug=None):
    """Solve triangular system of equations."""

def tensorsolve(a, b, axes=None):
    """Solve tensor equation a x = b for x."""

def inv(a):
    """Compute multiplicative inverse of matrix.
    
    Args:
        a: Matrix to invert
        
    Returns:
        cupy.ndarray: Inverse matrix
    """

def pinv(a, rcond=1e-15, hermitian=False):
    """Compute Moore-Penrose pseudo-inverse.
    
    Args:
        a: Matrix to pseudo-invert
        rcond: Cutoff for small singular values
        hermitian: Whether matrix is Hermitian
        
    Returns:
        cupy.ndarray: Pseudo-inverse matrix
    """

def tensorinv(a, ind=2):
    """Compute 'inverse' of N-dimensional array."""

def eigh(a, UPLO='L'):
    """Return eigenvalues and eigenvectors of Hermitian matrix.
    
    Args:
        a: Hermitian matrix
        UPLO: Whether to use upper ('U') or lower ('L') triangle
        
    Returns:
        tuple: (eigenvalues, eigenvectors)
    """

def eigvalsh(a, UPLO='L'):
    """Compute eigenvalues of Hermitian matrix."""

def svd(a, full_matrices=True, compute_uv=True):
    """Singular Value Decomposition.
    
    Args:
        a: Input matrix
        full_matrices: Whether to compute full or reduced SVD
        compute_uv: Whether to compute U and Vh matrices
        
    Returns:
        tuple: (U, s, Vh) or just s if compute_uv=False
    """

def qr(a, mode='reduced'):
    """Compute QR decomposition of matrix.
    
    Args:
        a: Input matrix
        mode: Decomposition mode ('reduced', 'complete', 'r', 'raw')
        
    Returns:
        tuple: (Q, R) matrices or modified based on mode
    """

def cholesky(a):
    """Cholesky decomposition of positive-definite matrix.
    
    Args:
        a: Positive-definite matrix
        
    Returns:
        cupy.ndarray: Lower triangular Cholesky factor
    """

def matrix_power(a, n):
    """Raise square matrix to integer power.
    
    Args:
        a: Square matrix
        n: Integer power
        
    Returns:
        cupy.ndarray: Matrix power a^n
    """

def cross(a, b, axisa=-1, axisb=-1, axisc=-1, axis=None):
    """Return cross product of two vectors.
    
    Args:
        a, b: Input vectors
        axisa, axisb, axisc: Axis specifications
        axis: Axis of vectors
        
    Returns:
        cupy.ndarray: Cross product
    """

def vdot(a, b):
    """Return dot product of two vectors."""

import cupy as cp

# Create matrices
A = cp.random.random((3, 3))
B = cp.random.random((3, 3))
x = cp.random.random(3)

# Matrix multiplication
C = cp.dot(A, B)          # Matrix-matrix multiplication
y = cp.dot(A, x)          # Matrix-vector multiplication
C_alt = cp.matmul(A, B)   # Alternative matrix multiplication

# Matrix properties
trace_A = cp.trace(A)     # Sum of diagonal elements
det_A = cp.det(A)         # Determinant
norm_A = cp.norm(A)       # Frobenius norm
norm_2 = cp.norm(A, ord=2) # 2-norm (spectral norm)

# Solve Ax = b
A = cp.array([[3, 1], [1, 2]], dtype=cp.float32)
b = cp.array([9, 8], dtype=cp.float32)

# Direct solution
x = cp.solve(A, b)  # x = [2, 3]

# Least squares solution for overdetermined systems
A_over = cp.random.random((10, 3))  # 10 equations, 3 unknowns
b_over = cp.random.random(10)
x_ls, residuals, rank, s = cp.lstsq(A_over, b_over)

# Matrix inversion
A_inv = cp.inv(A)
x_inv = cp.dot(A_inv, b)  # Alternative solution method

# Symmetric matrix eigenvalues
symmetric_matrix = cp.array([[4, -2], [-2, 1]], dtype=cp.float32)
eigenvals, eigenvecs = cp.eigh(symmetric_matrix)

# Note: CuPy only supports eigenvalue computation for Hermitian matrices
# For general matrices, use NumPy on CPU or specialized libraries

# SVD decomposition
matrix = cp.random.random((4, 3))
U, s, Vh = cp.svd(matrix, full_matrices=False)
reconstructed = cp.dot(U * s, Vh)  # Reconstruction

# QR decomposition
tall_matrix = cp.random.random((5, 3))
Q, R = cp.qr(tall_matrix)

# Cholesky decomposition for positive definite matrix
pos_def = cp.dot(matrix.T, matrix)  # Ensure positive definite
L = cp.cholesky(pos_def)

# Matrix functions
matrix = cp.array([[1, 2], [3, 4]], dtype=cp.float32)

# Matrix powers
matrix_squared = cp.matrix_power(matrix, 2)
# Vector operations
v1 = cp.array([1, 2, 3])
v2 = cp.array([4, 5, 6])

cross_product = cp.cross(v1, v2)  # Cross product
dot_product = cp.vdot(v1, v2)     # Dot product

# Multi-matrix dot product
matrices = [cp.random.random((3, 4)), 
           cp.random.random((4, 5)), 
           cp.random.random((5, 2))]
# Chain multiplication using matmul
result = cp.matmul(cp.matmul(matrices[0], matrices[1]), matrices[2])

# Use appropriate data types for better performance
A_f32 = cp.array(A, dtype=cp.float32)  # Single precision
A_f64 = cp.array(A, dtype=cp.float64)  # Double precision

# Prefer matmul over dot for matrix multiplication
result = cp.matmul(A, B)  # Preferred for matrix operations

# Use in-place operations when possible
cp.dot(A, B, out=C)  # Write result directly to C

# For large matrices, consider blocked algorithms
# CuPy automatically uses optimized BLAS routines

# Solving multiple systems with same coefficient matrix
A = cp.random.random((100, 100))
multiple_b = cp.random.random((100, 50))  # 50 different RHS vectors

# Efficient solution for multiple RHS
solutions = cp.solve(A, multiple_b)

# Rank analysis
rank = cp.matrix_rank(A)
if rank < A.shape[0]:
    print("Matrix is rank deficient")