Exam 8: Orthogonality

If S is a nonzero subspace of , then for every vector u in , belongs to S.

If is a nonzero subspace of , and u belongs to , then for every vector .

If is an invertible (square) matrix with singular value decomposition , then a singular value decomposition for is given by .

Find the QR decomposition for the matrix .

If is a QR factorization of a matrix A, then .

Find an orthogonal basis for the subspace .

Suppose and are orthogonal with and . Determine .

Find the singular values for the matrix .

If Q is an orthogonal matrix, then .

If is an orthogonal set of vectors, and , then .

If a system of equations has more equations than variables, then the system has a unique least squares solution.

Find the normal equations for the given system.

Determine if the vectors form an orthogonal set.

Determine from the given matrix with orthogonal columns without using row operations.

Find a singular value decomposition for the matrix .

The eigenvalues and corresponding eigenvectors for a symmetric matrix A are given. Find matrices D and P of an orthogonal diagonalization of A.

Find an orthonormal basis for the subspace .

Find a basis for for the subspace S.

If A is an diagonalizable matrix, then there exists a diagonal matrix D and an orthogonal matrix P such that .

Verify that the eigenvalues of are nonnegative.