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A complex Wishart matrix is the Gram matrix of an $n$-by-$m$ matrix of independent centered complex Gaussian random variables with an indicated variance convention. It is a positive semidefinite matrix. In the standard convention with joint entry density proportional to $e^{-\mathrm{Tr}{X}^{\ast }X}$ and $n\ge m$, its eigenvalue density contains the weight $x^{n-m}{e}^{-x}$ and a squared Vandermonde determinant.

The traceless symmetric part of the galaxy velocity-gradient matrix changes shape without isotropic expansion. An antisymmetric part instead represents rigid rotation. A matrix proportional to the identity plus an antisymmetric matrix has no shear.

For $\lambda >0$ and at least two distinct design points, the unique $C^2[a,b]$ minimizer of squared residuals plus $\lambda J(g)$ is $N(I+\lambda \Gamma {)}^{-1}\mathbf{Y}$. The minimum roughness property of the natural cubic spline interpolant reduces the optimization to the value vector. The matrix $I+\lambda \Gamma$ is a positive-definite matrix, so the normal equations have a unique solution; the interpolation equality case gives uniqueness as a function. Linear tails are part of the natural boundary convention.

If $f$ is a modular form on a finite-index subgroup and $\gamma$ is rational with positive determinant, then $f{|}_k\gamma$ is bounded near infinity. Choose $\sigma \in S{L}_2(\mathbb{Z})$ with $\sigma \infty =\gamma \infty$. The matrix ${\sigma }^{-1}\gamma$ is upper triangular and sends $z$ to $Az+B$ with $A>0$. Thus the existing cusp expansion of $f{|}_k\sigma$ stays bounded under this substitution. Rational conjugation of finite-index modular subgroups gives a positive period for the translate, so boundedness gives a removable singularity in its cusp parameter. Apply this to $\gamma \rho$ for every $\rho \in S{L}_2(\mathbb{Z})$ to obtain holomorphy at every cusp of the new subgroup. Vanishing at the cusps is preserved too.

For an invertible matrix $A$ and any matrix $H$,Indeed $\det (A+tH)=\det (A)\det (I+t{A}^{-1}H)$, and multilinearity shows that the linear coefficient of $\det (I+tB)$ is $\mathrm{tr}B$. At a singular matrix the derivative is instead written $\mathrm{tr}(\mathrm{adj}(A)H)$; this polynomial formula holds for all $A$.

For an integer matrix $M$, its $j$th determinantal divisor is the nonnegative greatest common divisor of all $j$-by-$j$ matrix minors, with ${\Delta }_0=1$. If all these minors vanish, take ${\Delta }_j=0$. Unimodular row and column operations preserve these divisors. For nonzero invariant factors $d_1\mid \cdots \mid d_r$ in the Smith normal form, ${\Delta }_j=d_1\cdots d_j$ for $j\le r$, so $d_j={\Delta }_j/{\Delta }_{j-1}$.

For a rational matrix $\gamma$ of positive determinant, use the positive real power of $\det \gamma$ in the displayed slash operator for modular forms. It satisfies $(f{|}_k\gamma ){|}_k\eta =f{|}_k(\gamma \eta )$ by the automorphy factor identity and multiplicativity of the determinant. It agrees with the usual operator on the modular group. The determinant factor is essential for the stated normalization of the Hecke operators.

Let $K_h$ be a symmetric nonnegative matrix and consider $U^{\prime \prime }=(\alpha I-K_h)U$ with fixed real $\alpha$. In any inner product for which $K_h$ is self-adjoint, diagonalization of a matrix givesIndeed each modal coefficient is either a cosine and sine divided by its frequency, a linear function at zero frequency, or a hyperbolic cosine and sine divided by its growth rate. The bounds $|\cos (\omega t)|\le 1$, $|\sin (\omega t)/\omega |\le t$, and $\sinh (\beta t)/\beta \le t{e}^{\beta t}$ prove the estimate. It is uniform in the spectrum of $K_h$ and therefore gives finite-time displacement stability. Uniform velocity estimates need additional control of the initial spatial energy. All-time displacement bounds require a strictly positive lower bound for $K_h-\alpha I$, a different condition.

For $r=|x|>0$, the gradient satisfies ${\partial }_{i}r=x_i/r$. The product rule and Einstein notation then give the displayed divergence formula in dimension $n$. In dimension three, its derivative matrix is symmetric, so the curl vanishes. Zero divergence forces $f(r)=C{r}^{-n}$ on each positive radial interval. This formula does not assert regularity at the origin.

A square real matrix is doubly stochastic if its entries are nonnegative and every row and column sums to $1$. Its positive-entry support satisfies the condition of the Hall marriage theorem: for a row subset $S$, $|S|\le |N(S)|$ by summing column capacities. Consequently its support contains a permutation matrix.

Geometric multiplicity is the dimension of an eigenspace; algebraic multiplicity is its multiplicity as a root of the characteristic polynomial. For a real symmetric matrix they coincide because it has an orthonormal eigenbasis.

A nonlinear term $f(|x|)Nx$ with real skew-symmetric matrix $N$ contributes zero to the derivative of the quadratic energy $|x{|}^2/2$. It changes orientation rather than magnitude directly. If a separate linear generator is a non-normal matrix, that reorientation can still alter later transient growth, so energy cancellation alone does not make the nonlinear evolution equivalent to the linear one.

Applying a scalar function separately to each matrix entry: $(f[X]{)}_{ij}=f(X_{ij})$. This differs from spectral matrix functional calculus. In particular, $\arcsin [Y]$ in Gaussian hyperplane rounding is entrywise, and its contribution to a matrix trace can be restricted to selected blocks by the support of the other matrix.