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A Hermitian form is positive semidefinite when $h(v,v)\ge 0$ for every vector. It need not define an inner product, because a nonzero vector may have zero squared norm. The Cauchy-Schwarz inequality still holds: applying nonnegativity to $v+zw$ and minimizing the quadratic expression in $z$ gives $|h(v,w){|}^2\le h(v,v)h(w,w)$ when $h(w,w)>0$; when $h(w,w)=0$, varying $z$ forces $h(v,w)=0$. Thus its zero-norm vectors are exactly the radical of a Hermitian form. Quotienting this radical of a Hermitian form gives a positive inner product; taking its Hilbert space completion then gives a Hilbert space. This is the final positivity step in the Gupta-Bleuler null-state quotient.

A real symmetric or complex Hermitian matrix $A$ is positive semidefinite when $x^{†}Ax\ge 0$ for every vector $x$, equivalently when all its eigenvalues are nonnegative.

Iterate $q^{(k+1)}=A{q}^{(k)}/\Vert A{q}^{(k)}\Vert$. With a unique dominant eigenvalue separated by a spectral gap and a starting vector that is nonorthogonal to its eigenvector, the directions converge at the spectral-ratio rate.

An axial vector transforms as ${\mathbf{a}}^{\prime }=(\det Q)Q\mathbf{a}$ under an orthogonal matrix $Q$. It agrees with a polar vector under a proper rotation matrix and has the additional determinant sign under an orientation-reversing transformation. Angular momentum and angular velocity are axial vectors.

Suppose an array $T_{ij}$ is specified in each right-handed orthonormal basis. If $w_i=T_{ij}{v}_j$ transforms as a vector for every vector $v$, then $T$ is a second-order Cartesian tensor under those basis changes. Indeed $v^{\prime }=Rv$, $w^{\prime }=Rw$ imply $T^{\prime }Rv=RTv$ for every $v$, hence $T^{\prime }=RT{R}^T$. The condition on every test vector is essential; one contraction is insufficient.

The radical of a Hermitian form is the vector subspace orthogonal to the entire space. The form descends to the quotient vector space by its radical: adding a radical vector to either argument does not change the value. For a positive semidefinite Hermitian form, the Cauchy-Schwarz inequality identifies this radical with its zero-norm vectors, so the quotient form is positive definite. This conclusion does not hold for a general indefinite Hermitian form: a zero-norm vector need not belong to its radical.

For real symmetric $A$, replace sign-vector lifts $x{x}^T$ by all matrices in the elliptope:Every sign vector gives a feasible rank-one matrix with the same objective, so the relaxed maximum is an upper bound. The omitted rank-one condition is substantive: an arbitrary feasible Gram matrix need not come from signs.

With metric signature $(-,+,+,+)$, a nonzero vector $v$ is spacelike when its Lorentzian inner product with itself satisfies $g(v,v)>0$. In Minkowski spacetime, this means its spatial components have larger squared Euclidean length than its time component. The classification is preserved by Lorentz transformations. A timelike vector has negative squared Lorentzian inner product, and a null vector has zero squared Lorentzian inner product.

A linear map is injective on the class of sparse vectors of order $s$ exactly when its null space contains no nonzero vector with at most $2s$ nonzero entries. Differences of two sparse vectors have at most $2s$ active coordinates. Conversely, splitting the support of a vector of a $2s$-sparse null vector into two parts constructs two distinct $s$-sparse vectors with the same image. The strict null space property implies sparse injectivity by applying it to both parts of such a split.

For a design matrix $X$ with $p$ columns, define ${\kappa }_m^2={\max }_{|M|=m}{\lambda }_{\max }(X_M^T{X}_M/n)$ for $1\le m\le p$. Equivalently this bounds $\Vert Xv{\Vert }_2^2/(n\Vert v{\Vert }_2^2)$ for vectors supported on at most $m$ coordinates. These quantities are nondecreasing: extend the support of a maximizing vector by zero coordinates and use the variational characterization of the largest eigenvalue. Values at $m>p$, including infinity, require a separate convention.

For a Dirac spinor a spatial rotation acts in each chiral block by a matrix exponential $e^{\pm i\theta \mathbf{n}\cdot \mathbf{\sigma }/2}$, depending on angle orientation. Since $(\mathbf{n}\cdot \mathbf{\sigma }{)}^2=I_2$ for a unit axis, the Pauli matrix multiplication law gives this exponential as $\cos (\theta /2)I_2\pm i\sin (\theta /2)\mathbf{n}\cdot \mathbf{\sigma }$. At $\theta =2\pi$ it is $-I_2$, and at $4\pi$ it is $I_2$. A vector returns already at $2\pi$. This describes the lift of the rotation loop to the Spin group, rather than a single-valued representation of the ordinary rotation group on spinors.