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Subsection 2.1.3 What you will learn

This week introduces two concepts that have theoretical and practical importance: unitary matrices and the Singular Value Decomposition (SVD).

Upon completion of this week, you should be able to

  • Determine whether vectors are orthogonal.

  • Compute the component of a vector in the direction of another vector.

  • Relate sets of orthogonal vectors to orthogonal and unitary matrices.

  • Connect unitary matrices to the changing of orthonormal basis.

  • Identify transformations that can be represented by unitary matrices.

  • Prove that multiplying with unitary matrices does not amplify relative error.

  • Use norms to quantify the conditioning of solving linear systems.

  • Prove and interpret the Singular Value Decomposition.

  • Link the Reduced Singular Value Decomposition to the rank of the matrix and determine the best rank-k approximation to a matrix.

  • Determine whether a matrix is close to being nonsingular by relating the Singular Value Decomposition to the condition number.