![[Pasted image 20260519093612.png]] ![[Pasted image 20260519124224.png]] Matrix types: - Upper Triangular: $A_{i,j}=0$ for all $i<j$ - Lower Triangular: $A_{i,j}=0$ for all $i>j$ - Symmetric: $S = S^{\mathrm{T}}$ - Invertible - Square matrix $A$ is invertible if and only if: - $\det(A) \neq 0$ - all $\lambda \neq 0$ - Singular - Square matrix $A$ is singular if and only if: - at least one $\lambda=0$ - $\det(A)=0$ - Permutation - square with eactly one 1 in each row and column, and the rest is zeros - if A is a permutation matrix: - $A^{\mathrm{T}}= A^{-1}$ - All $\lambda$ are roots of 1 - Doesn't change the length of a vector - Doesn't change the angle between vectors - Positive Definite - $x^{\mathrm{T}} Ax > 0$ for all $x$ - All $\lambda>0$ - Positive Semidefinite - A is PSD if and only if: - $x^{\mathrm{T}} Ax \geq 0$ for all $x$ - all $\lambda \geq 0$ - $f(x) = x^{\mathrm{T}}Ax$ is convex - $A = B^{\mathrm{T}}B$ for some $B$ - All PSD matrices are symmetric - If A and B are PSD, so is $A+B$ - If B is PSD, so is $B^{\mathrm{T}}AB$ - Unitary - Complex equivalent of orthogonal; $U$ is unitary if and only if $U^{*}U=UU^{*}=I$ - Referred to as $U^{\dagger}$ and the "hermitian adjoint" of a matrix in quantum - Markov - nonnegative matrix where each row or column sums to 1 - if rows sum to 1, it's called row stochastic, otherwise column stochastic - it has an eigenvalue 1 with a corresponding nonnegative eigenvector - other eigenvalues have absolute values $\leq 1$ ### Matrix Shapes The following statements are equivalent: | Square | Tall | Wide | | ----------------------------------------- | ------------------------------------------ | ------------------------------------- | | Linearly independent rows and columns | Linearly independant columns | Linearly independant rows | | $Ax = b$ is uniquely solvable for any $b$ | $Ax = b$ has a unique solution if solvable | $Ax = b$ is solvable for any real $b$ | | Invertible | Has left inverse | Has a right inverse | | ${} N(A)=0 {}$ | ${} N(A)=0 {}$ | | | ${} C(A)=\mathbb{R}^{n} {}$ | | ${} C(A) = \mathbb{R}^{m} {}$ | | ${} \text{Rank}(A)=m=n {}$ | ${} \text{Rank}(A)=n {}$ | $\text{Rank}(A)=m$ |