Inner products, spectral reasoning and matrix structure
A matrix calculation should begin with the field, dimensions and the transformation represented. Rank-nullity relates the image and kernel of a finite-dimensional linear map, while an invertible change of basis changes coordinates without changing the underlying map. Distinguish similarity, which represents the same endomorphism in different bases, from congruence, which transforms a bilinear or quadratic form. A matrix can have repeated eigenvalues without a full eigenbasis. The algebraic and geometric multiplicities answer different questions.
Inner products supply norms, orthogonality and projection. In a real inner-product space, equality in Cauchy-Schwarz holds when the vectors are linearly dependent, including the zero-vector cases. To project onto a finite-dimensional subspace, require the residual to be orthogonal to every vector in a basis of that subspace. An orthonormal basis simplifies the coefficients; otherwise solve the Gram system. For polynomial functions with an integral inner product, parity can make cross terms zero. Orthogonality is with respect to the stated inner product, not necessarily the unweighted coordinate dot product.
Real symmetric and complex Hermitian matrices have orthonormal eigenbases with real eigenvalues. Positive definiteness can be checked through positive eigenvalues or appropriate factorization and minor criteria under their actual hypotheses. A positive-definite matrix defines an inner product. Simultaneous reduction of a positive-definite form and another symmetric form is a congruence result; it does not claim that arbitrary matrices can always be simultaneously diagonalized by similarity. State which transformation is being sought before selecting a theorem.
Operator norms and eigenvalues are related but distinct. The Euclidean operator norm is the largest singular value and equals the square root of the largest eigenvalue of A* A. Spectral radius is at most any consistent subordinate operator norm. Normal matrices have Euclidean norm equal to spectral radius, but equality alone does not characterize normality in every dimension. For example, a block with a dominant scalar entry can determine both quantities while another smaller block is non-normal. A necessary-versus-sufficient distinction can invalidate an apparently familiar multiple-choice statement.
For bounded operators on an inner-product space, the orthogonal complement of the image equals the kernel of the adjoint. In infinite-dimensional settings, image closure and compactness can matter. Compact self-adjoint operators have real nonzero eigenvalues with finite-dimensional corresponding eigenspaces and possible accumulation only at zero; a complete description must also account for the kernel. Do not transfer every finite-matrix assertion to an arbitrary infinite-dimensional operator. In applied problems, connect eigenvalues, quadratic forms, linear systems and projections with the actual assumptions rather than selecting a theorem from one keyword.