## Transpose Definition: $(A^{\mathrm{T}})_{i,j}=A_{j,i}$ $A$ is symmetric if $A=A^{\mathrm{T}}$ $(AB)^{\mathrm{T}}=B^{\mathrm{T}}A^{\mathrm{T}}$ $(A^{\mathrm{T}})^{\mathrm{T}}=A$ $(A+B)^{\mathrm{T}}=A^{\mathrm{T}}+B^{\mathrm{T}}$ For square, non-singular A we have $(A^{-1})^{\mathrm{T}}=(A^{\mathrm{T}})^{-1}$ ## Conjugate Transpose Same as transpose except applies the conjugate transpose ($\overline{a+bi}=a-bi$) to each complex entry $A^{\star}=\overline{A^{\mathrm{T}}}$