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For a barotropic fluid, the squared sound speed is $c_s^2=\partial P/\partial \rho$. A Fourier density mode obeysin linear Newtonian perturbation theory.

The Jeans wavenumber separates pressure-supported modes from gravitationally unstable density modes.

The density contrast is the fractional perturbation $\delta =(\rho -\bar{\rho })/\bar{\rho }$.

An upper or lower triangular matrix has all entries on one side of the main diagonal equal to zero. Its eigenvalues, with algebraic multiplicity, are its diagonal entries.

A diagonal matrix has zero entries away from its main diagonal. Its eigenvalues are its diagonal entries.

For $M⪰0$ with $\mathrm{tr}M\le s$, $\mathrm{tr}(MA)\le s\max ({\lambda }_{\max }(A),0)\le s\Vert A{\Vert }_{\mathrm{op}}$.

The tensor product of linear maps is determined by $(A\otimes B)(x\otimes y)=Ax\otimes By$ and linear extension.

The real orthogonal group is $O(n)=\{R:R^{T}R=I\}$. Its tangent space at the identity is the vector space of skew-symmetric matrices.

The entrywise product of two positive semidefinite matrices is positive semidefinite. Writing each matrix as a sum of rank-one Gram matrices reduces the claim to $(u{u}^T)\circ (v{v}^T)=(u\circ v)(u\circ v{)}^T$.

A quadratic form on a vector space over a field $F$ is a function $Q:V\to F$ satisfying $Q(av)=a^{2}Q(v)$ whose polar bilinear form of a quadratic form $B_Q(v,w)=Q(v+w)-Q(v)-Q(w)$ is a bilinear form. In finite dimension, it is represented in coordinates by a homogeneous degree-two polynomial, or by $Q(v)=v^{T}Av$ with a suitable matrix $A$.

When the characteristic is not two, one may choose $A$ to be a symmetric matrix and recover it from $B_Q/2$. In characteristic two, the symmetric-matrix restriction would lose examples: $Q(x,y)=xy$ has an upper-triangular representing matrix $\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$, while $v^{T}Av$ for a symmetric matrix has no $xy$ cross term. Its polar form is $B_Q((x,y),(u,v))=xv+yu$, an alternating bilinear form. The distinction matters when studying quadratic forms over finite fields.^[ref]

An inner product assigns a scalar $⟨u,v⟩$ to two vectors and supplies their lengths and orthogonality. Using the convention linear in the first argument, it satisfiesThe resulting norm is $\Vert v\Vert =\sqrt{⟨v,v⟩}$. Over the real numbers, conjugation has no effect and the form is symmetric and bilinear. Over the complex numbers, it is sesquilinear; the convention linear in the second argument is also common, so formulas for coefficients must match the stated convention.

The Euclidean example is $⟨u,v⟩={\sum }_i{u}_i\overline{v_i}$. For real $u=(2,1)$ and $v=(1,3)$ this gives $⟨u,v⟩=5$, while $\Vert u{\Vert }^2=5$ and $\Vert v{\Vert }^2=10$. If $r=(-1,2)$, then $⟨r,u⟩=0$: perpendicularity becomes an algebraic equation. The Cauchy-Schwarz inequality gives $|⟨u,v⟩|\le \Vert u\Vert \Vert v\Vert$, and in a real inner product space it makes the angle formula $\cos \theta =⟨u,v⟩/(\Vert u\Vert \Vert v\Vert )$ meaningful for nonzero vectors.

Different inner products express different measurements. The weighted example $⟨u,v⟩=4u_1{v}_1+u_2{v}_2$ on ${\mathbb{R}}^2$ gives $\Vert (x,y){\Vert }^2=4x^2+y^2$, so its unit-length vectors lie on an ellipse. For functions, an integral such as $⟨f,g⟩=\int f\overline{g}$ supplies the geometry of the L2 space. This lets orthogonal projection and least-squares solutions apply to curves as well as coordinate lists.

Every complex square matrix is a product of two symmetric matrices, one invertible.

Orthogonal similarity maps A to QAQ^T and preserves eigenvalues and symmetry.

For a block matrix $M=\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)$ with $A$ an invertible matrix, the Schur complement of $A$ in $M$ is $S=D-C{A}^{-1}B$. It is the effective coefficient matrix for the remaining variables after block elimination. The definition requires invertibility of $A$, but not of $S$ or $M$. Its size is that of $D$, even when $B$ and $C$ are rectangular matrices.

In the original example under block matrix inverse, $D=5$ becomes $S=19/6$ because changing the remaining variable also changes the two variables solved through $A$. This is what makes a Schur complement more informative than simply deleting rows and columns. It retains their mediated effect, and its matrix inverse is the corresponding block of $M^{-1}$ when the block matrix inverse exists.

The same elimination appears when minimizing a quadratic form. The Schur complement as a minimized quadratic form gives a geometric example; the Schur complement determinant identity gives an exact test for an invertible matrix.

For maps through a finite-dimensional intermediate space V, rank(alpha beta) is at least rank(alpha)+rank(beta)-dim(V).

Equivalent matrices represent the same linear map after independent changes of domain and codomain bases.

The adjugate is the transpose of the cofactor matrix and satisfies $A\mathrm{adj}(A)=(\det A)I$.

For invertible $I-A$, ${\sum }_{j=0}^{n-1}{A}^j=(I-A^n)(I-A{)}^{-1}$.

A Cartesian coordinate system specifies a point by its signed components along mutually perpendicular coordinate axes.

A rotation matrix is an orthogonal matrix with determinant $1$.

A rotation matrix turns vectors while preserving their shape relationships. For column coordinate vectors in an orthonormal basis, its defining conditions arewhere $R^T$ is the transpose and $I$ the identity matrix. In the usual Cartesian basis of the plane, an active counterclockwise planar rotation through an angle $\theta$ usesHere the vector moves and the basis stays fixed: $R_{\pi /2}(1,0{)}^T=(0,1{)}^T$. Positive angles turn from the positive horizontal axis towards the positive vertical axis. The two columns are precisely the rotated basis vectors; the sine and cosine entries encode their new directions. Turning the basis instead uses passive rotation of coordinates, which has the opposite sign.

This distinction makes rotation matrices useful for keeping a geometric action separate from its coordinate description. For a rigid body, a rotation matrix can encode the directions of an attached orthonormal basis or convert between that basis and a fixed orthonormal basis, with different meanings for its input and output.^[ref]

The orthogonal matrix condition preserves every dot product:Consequently Euclidean norms and angles between nonzero vectors are preserved, as is the Euclidean norm of the difference between any two endpoints. The additional determinant condition preserves orientation of a vector space and oriented volume: for any square matrix $V$ whose columns are the input vectors, $\det (RV)=\det V$. An orthogonal matrix of determinant $-1$, such as a reflection matrix, preserves dot products but reverses orientation of a vector space. The matrix inverse is especially simple: $R^{-1}=R^T$.^[ref]