State definitions of the following.

• Characteristic equation of a matrix.
• Eigenvalue, eigenvector.
• Similar matrices.
• Diagonalizable matrix.
• Eigenvector basis.
• Attractor, repellor, saddle point of a dynamical system.
• Exponential of a matrix.
• Standard inner product in Rⁿ.
• Transpose of a vector, transpose of a matrix.
• Norm of a vector in Rⁿ.
• Distance and angle between vectors in Rⁿ.
• Orthogonal vectors.
• Orthogonal complement of a subspace.
• Orthogonal basis of a subspace; orthonormal basis of a subspace.
• Orthogonal projection on a subspace.
• Least squares solution of Ax=b, normal equations.

Do the following.

• Evaluate the characteristic polynomial of a matrix.
• Find the eigenvalues of a matrix, and a basis for the eigenspace belonging to each eigenvalue.
• If A diagonalizable, find invertible P and diagonal D so that A=PDP^-1. Find a formula for A^k.
• Find a factorization of a real 2×2 matrix that exhibits the rotation associated with a complex eigenvalue.
• Make a change of variable to decouple a dynamical system x(t+1)=A x(t) or x'(t)=A x(t) having state vector x in R².
• Calculate the inner product of vectors, the norm of a vector, and the angle between vectors.
• Find the orthogonal projection of a vector on a subspace. Decompose a vector into the sum of a vector in a subspace W and a vector in W^perp.
• Use the Gram-Schmidt algorithm to find an orthonormal basis for the span of a set of linearly independent vectors.
• Find a QR factorization of a matrix with linearly independent columns.
• Use the normal equations to find the least squares solution of a system A x = b with a matrix A of full rank.
• Use the QR factorization to find the least squares solution of a system A x = b with a matrix A of full rank.
• Use the relations among the four fundamental subspaces of a matrix.

alex at iastate dot edu

Document last modified on Thu Aug 10 2006