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Orthogonality

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Definition of angle between vectors
Definition of orthogonal vectors
Definition of parallel vectors
Pythagorean equality characterizes orthogonality over the reals
Definition of a vector being orthogonal to a subspace
Definition of orthogonal complement of a subspace
The orthogonal complement of a subspace is a subspace.
The direct sum of a subspace and its orthogonal complement is the whole space.
A vector is in the orthogonal complement of a subspace if and only if it is orthogonal to every vector in a basis of the subspace.
Null spaces are orthogonal complements of the corresponding row or column spaces
Definition of orthogonal set of vectors
Definition of orthonormal set of vectors
An orthogonal set of nonzero vectors is linearly independent.
Definition of orthogonal basis of a (sub)space
Definition of orthonormal basis of a (sub)space
A square real matrix has orthonormal columns iff its inverse is its transpose
A matrix with real entries and orthonormal columns preserves norms.
A matrix with real entries and orthonormal columns preserves dot products.
Formula for the coordinates of a vector with respect to an orthogonal/orthonormal basis.
A vector can be written uniquely as a sum of a vector in a subspace and a vector orthogonal to the subspace.
Description of the Gram-Schmidt process
The Gram-Schmidt process converts a linearly independent set into an orthogonal set.
Definition of Gram-Schmidt process
Definition of orthogonal subspaces

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A square real matrix has orthonormal columns iff its inverse is its transpose

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A matrix with real entries and orthonormal columns preserves norms.Close match
Definition of orthonormal set of vectorsClose match
A matrix with real entries and orthonormal columns preserves dot products.Close match
Definition of orthogonal matrixClose match
Unitary matrices have orthogonal (orthonormal) rows/columns.Close match
If a square matrix has a one-sided inverseClose match
Equivalence theorem for nonsingular matrices: the transpose of the matrix A has an inverse.Close match
Equivalence theorem for nonsingular matrices: the matrix A has a right inverse.Close match
Equivalence theorem for nonsingular matrices: the matrix A has a left inverse.Close match
Equivalence theorem for nonsingular matrices: the matrix A has an inverse.Close match
Definition of inverse of a matrixClose match
The QR decomposition of a nonsingular matrix exists.Close match
Matrix transpose commutes with matrix inverse.Close match
Matrix inverses are unique: if A and B are square matricesClose match
A matrix and its transpose have the same determinant.Close match
Matrix transpose is an involution.Close match
A matrix and its transpose have the same eigenvalues/characteristic polynomial.Close match
A matrix is orthogonally diagonalizable if and only if it is symmetric.Close match
The spectral theorem for symmetric matricesClose match
Definition of orthogonally diagonalizable matrixClose match

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