A matrix is a rectangular array of numbers. Its transpose \(A^T\) is obtained by swapping rows and columns, so \((A^T)_{ij} = A_{ji}\). The inverse \(A^{-1}\) of a square matrix is the unique matrix satisfying \(AA^{-1} = A^{-1}A = I\), where the identity matrix \(I\) has 1s on the diagonal and 0s elsewhere. Inversion is possible exactly when \(\det(A) \neq 0\).
Many special matrices arise repeatedly. A diagonal matrix has nonzero entries only on the main diagonal, while an upper (or lower) triangular matrix has zero entries below (or above) the diagonal. A symmetric matrix satisfies \(A = A^T\), and a skew-symmetric matrix satisfies \(A^T = -A\), forcing zero diagonal entries. An orthogonal matrix \(Q\) satisfies \(Q^T Q = Q Q^T = I\), meaning its columns form an orthonormal set. The complex analogue is the unitary matrix, where \(U^*U = UU^* = I\) with \(U^*\) denoting the conjugate transpose. A matrix is normal when \(AA^* = A^*A\); normal matrices are precisely those that can be unitarily diagonalized. A Hermitian matrix is the complex analogue of a symmetric one, equal to its own conjugate transpose. A symmetric matrix is positive definite if \(x^T A x > 0\) for every nonzero real vector \(x\), and positive semidefinite when the inequality is non-strict; Sylvester's criterion gives an alternative test by requiring all leading principal minors to be positive.
A linear transformation is a function \(T: V \to W\) between vector spaces that respects the structure: \(T(u+v) = T(u) + T(v)\) and \(T(cv) = c T(v)\). Once bases are chosen for \(V\) and \(W\), \(T\) is represented by a matrix \(A\) with \(T(x) = Ax\); the columns of \(A\) are the images of the basis vectors of \(V\). The product of two matrices corresponds to the composition of the corresponding linear maps. Two matrices \(A\) and \(B\) are similar when \(A = P^{-1} B P\) for some invertible \(P\); similar matrices represent the same linear map in different bases, and they share the same characteristic polynomial, trace, determinant, rank, and eigenvalues. The change-of-basis matrix from basis \(B\) to basis \(C\) has columns equal to the \(C\)-coordinates of the \(B\) basis vectors, converting \(B\)-coordinates into \(C\)-coordinates.