Anki cards for the second year of maths at Cambridge, quality varies drastically by module
| Text | Definition. For \(A \in \text{Mat}_n(\mathbb{F})\) define the trace of A {{c1:: \[\text{tr}(A) = \sum_{i=1}^n a_{ii}\]}} also define the characteristic polynomial of A as {{c1::\[\mathcal{X}_A (t) = \det (t \cdot I_n - A)\]}} |
| Back Extra |
| Front | Theorem. Let \(\mathbf{x}\) be a basic feasible solution associated with a basis matrix B and let \(\mathbf{\overline{c}}\) be the vector of reduced costs. Then \(\mathbf{x}\) is optimal if and only if \(\mathbf{\overline{c}} \geq 0\). |
| Back | The optimality conditions are primal feasibility, dual feasibility and complementary slackness.If \(\mathbf{x}\) is a basic feasible solution, we can substitute this into the complementary slackness equation and pick \(\bf \lambda\) such that it holds. Then for dual feasibility to hold we need the vector of reduced costs - \(\mathbf{\overline{c}} = \mathbf{c} - A^T(B^T)^{-1} \mathbf{c_b}\) - to be non-negative. If this has any negative elements, we can move in the jth direction and decrease cost as going from \(\mathbf{x} \mapsto \mathbf{x} + \theta \mathbf{d}\) we get \(\mathbf{c^T(x + \theta d)} = \mathbf{c^T x} + \theta \overline{c_j}\). |
| Front | Definition. What is a stochastic matrix? |
| Back | When the sum of each row of the transition matrix is equal to 1.\[\sum_{y \in I} P(x,y) = 1\] |