Matrix logarithm ($\log A$)

For a positive-definite Hermitian matrix with spectral decomposition $A=U\mathrm{diag}({\lambda }_j)U^{\ast }$, the matrix logarithm is $\log A=U\mathrm{diag}(\log {\lambda }_j)U^{\ast }$. More generally it is a branch of the inverse of the matrix exponential when such a branch exists.

On complex symmetric matrices with positive-definite real part, the determinant has a nonzero analytic square root normalized positively on real positive matrices. It is continued within this accretive domain. For $A=G+iB$, put $C=G^{-1/2}B{G}^{-1/2}$ with real eigenvalues $b_j$; then $d(A)=\sqrt{\det G}{\prod }_j\sqrt{1+i{b}_j}$, using positive-real-part roots for each factor. The principal scalar square root of the product determinant need not equal this continued root: accumulated determinant arguments can cross the scalar branch cut. The principal matrix logarithm instead tracks all factors consistently.

For a positive-definite Hermitian operator $A={\sum }_j{t}_j|u_j⟩⟨u_j|$ and a unit vector $v$, the weights $|⟨v|u_j⟩{|}^2$ form a probability distribution. Scalar logarithmic concavity bounds their average of $\log t_j$ by the logarithm of their average of $t_j$. Positive semidefinite operators follow by a limiting convention. Using the eigenbasis of a density operator $\rho$ gives $S(\rho \Vert \sigma )\ge D(r\Vert q)$, where $r$ is the spectrum of $\rho$ and $q$ the diagonal of $\sigma$ in that basis. This yields nonnegativity of quantum relative entropy without commutativity.

An invertible complex Jordan block $\lambda I+N$, with $N^s=0$, has logarithm $\ell I+{\sum }_{k=1}^{s-1}(-1{)}^{k+1}(N/\lambda {)}^k/k$, where $e^{\ell }=\lambda$. Applying this construction blockwise and conjugating back proves surjectivity of the matrix exponential on the complex general linear group. There is no globally single-valued continuous choice of logarithm.

For positive-definite Hermitian matrices $A,B$, Klein's inequality giveswith equality exactly when $A=B$. To prove it, let $a_i,b_j$ be their eigenvalues and $u_i,v_j$ their orthonormal eigenvectors. The weights $w_{ij}=|⟨u_i,v_j⟩{|}^2$ have row and column sums one. The difference between the two sides isby the scalar logarithm inequality $\ln t\le t-1$. Equality requires $a_i=b_j$ whenever $w_{ij}>0$, implying $A=B$. Limits extend the result to positive semidefinite matrices with the appropriate support condition. Applying it to trace-one matrices proves nonnegativity of quantum relative entropy.