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Linear Algebra - Matrix Operations
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LU decomposition:
L = \begin{bmatrix} 1 & 0 \\ 1.5 & 1 \end{bmatrix}, U = \begin{bmatrix} 4 & 3 \\ 0 & -1.5 \end{bmatrix}, A = LU
Element-wise multiplication (Hadamard product): ,
Subtract two matrices: ,
Matrix-vector multiplication: ,
Inverse of a 2x2 matrix:
Transpose of a matrix:
Rank of a matrix:
Eigenvalues of a matrix:
Eigenvalues are and
Solve a system of linear equations using matrix inverse:
Find the trace of a matrix:
Matrix power: ,
Matrix norm: where and using the Frobenius norm
Determinant of a 2x2 matrix:
Multiply a matrix by a scalar:
Matrix multiplication: ,
Cramer's Rule to solve for where and
Diagonalize a matrix:
Add two matrices: ,
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