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A square matrix is diagonalisable when it is similar to a diagonal matrix, that is, when its space has a basis of eigenvectors.

The adjugate of a square matrix is the transpose of its matrix of cofactors.

A linear form on a vector space is a linear map from the space to its field of scalars.

A permutation matrix is a square matrix with exactly one entry $1$ in each row and each column, and all other entries $0$. It permutes the coordinates of a vector.

An endomorphism of a finite-dimensional vector space is semisimple when each invariant subspace has an invariant complement. Over an algebraically closed field, this means that it is diagonalisable.

A subspace is invariant under a linear map from a space to itself when the map sends every vector of the subspace into the subspace.

The generalised eigenspace of an endomorphism for a scalar $\lambda$ is a set of vectors. It contains the vectors that some power of the map minus $\lambda$ times the identity sends to zero.

A unitary matrix is a complex square matrix whose conjugate transpose is its inverse.

A block matrix is a matrix that is divided into smaller matrices, its blocks, by horizontal and vertical lines.

The annihilator of a subspace is the set of linear forms that vanish on every vector of the subspace.

A symmetric bilinear form is a bilinear form that does not change when its two arguments are swapped.

The dual map of a linear map $T \colon V \to W$ is the linear map $W^* \to V^*$ that sends each linear form $f$ to $f \circ T$.

The orthogonal projection onto a subspace $W$ sends each vector to the unique vector of $W$ whose difference with it is orthogonal to $W$.

A linear isomorphism is a bijective linear map. Its inverse is also linear, and two spaces with an isomorphism between them are isomorphic.

The tensor product of two vector spaces is a vector space spanned by symbols $v \otimes w$. These symbols are bilinear in $v$ and $w$, and no other relations hold.

The quotient space of a vector space by a subspace is the set of cosets of the subspace, with the operations inherited from the space. It identifies two vectors when their difference lies in the subspace.

An alternating bilinear form is a bilinear form whose value on each pair of equal vectors is zero.

The linear span of a set of vectors is the set of all linear combinations of the vectors. It is the smallest subspace that contains the set.

An invertible matrix is a square matrix that has a two-sided inverse for matrix multiplication.

The product of two vector spaces over the same field is the set of pairs, with addition and scalar multiplication done in each component.

The eigenspace of a linear map for a scalar $\lambda$ is the kernel of the map minus $\lambda$ times the identity. It consists of the eigenvectors for $\lambda$ together with zero.

The coefficient matrix of a system of linear equations is the matrix of the coefficients of the unknowns. Its row $i$ holds the coefficients of equation $i$.

An endomorphism of a vector space is a linear map from the space to itself.

The sum of two subspaces is the set of all sums of a vector of the first subspace and a vector of the second subspace. It is the smallest subspace that contains both.

A square matrix is a matrix with the same number of rows and columns.

An alternating multilinear map is a multilinear map that is zero whenever two of its arguments are equal.

An idempotent matrix is a square matrix that is equal to its own square.

A basis of a vector space is a linearly independent family of vectors that spans the space. Each vector of the space is then a linear combination of the basis vectors in exactly one way.

Let $Q$ be a quadratic form over a field of characteristic not two. The symmetric bilinear form associated with $Q$ is the unique symmetric bilinear form $B$ with $B(v, v) = Q(v)$.

A positive definite matrix is a Hermitian matrix $A$ with $x^* A x > 0$ for every nonzero vector $x$.

The change of basis matrix from one basis to another is a square matrix. Its columns are the coordinate vectors of the new basis vectors in the old basis. It converts new coordinates into old coordinates.

A normal matrix is a complex square matrix that commutes with its conjugate transpose.

The quotient map of a vector space onto a quotient space sends each vector to its coset. It is a surjective linear map whose kernel is the subspace.

An upper triangular matrix is a square matrix whose entries below the main diagonal are all zero.

An eigenvector of a linear map from a space to itself is a nonzero vector that the map sends to a scalar multiple of itself.

A Hermitian matrix is a complex square matrix that is equal to its conjugate transpose.

A skew-symmetric matrix is a square matrix whose transpose is its negative.

The null space of a matrix is the set of vectors that the matrix sends to zero. It is the solution set of the homogeneous system with this matrix.

An inner product space is a real or complex vector space together with an inner product.

The adjoint of a linear map between finite-dimensional inner product spaces is a linear map in the opposite direction. It moves the map from one side of the inner product to the other.