Problem 1
Define by . Find a basis for with the property that is diagonal.
(a).
(b).
(c).
(d).
Problem 2
Let and , for , . Define by .
(a). Verify that is an Eigenvector of but is not Diagonalizable
(b). Find the -matrix for
Problem 3
Define by , where is a matrix with eigenvalues and . Does there exist a basis for such that the -matrix for is a diagonal matrix? Discuss.
Problem 4
Assume the following matrices are square.
(a). If is Invertible and Similar to , then is Invertible and is Similar to . Hint: for Some Invertible . Explain why is Invertible. then Find an Invertible such that
(b). If is Similar to , then is Similar to
(c). If is Similar to and is Similar to , then is Similar to
(d). If is Diagonalizable and is Similar to , then is also Diagonalizable
(e). If and is an Eigenvector of Corresponding to an Eigenvalue , then is an Eigenvector of Corresponding also to
(f). If and Are Similar, then They Have the Same Rank. Hint: Refer to Supplementary Exercises 13 and 14 for Chapter 4
Problem 5
The trace of a square matrix is the sum of the diagonal entries in and is denoted by . It can be verified that for any two matrices and . Show that if and are similar, then .
Problem 6
It can be shown that the trace of a matrix equals the sum of the eigenvalues of . Verify this statement for the case when is diagonalizable.
Problem 7
Let be with a basis ; let be with the standard basis, denoted here by ; and consider the identity transformation , where . Find the matrix for relative to and . What was this matrix called in Section 4.4?
Problem 8
Let be a vector space with a basis , be the same space with a basis , and be the identity transformation . Find the matrix for relative to and . What was this matrix called in Section 4.7?
Problem 9
Let be a vector space with a basis . Find the -matrix for the identity transformation .
Problem 10
Find the -matrix for the transformation when .
(a).
(b).
Problem 11
Let be the transformation whose standard matrix is given below. Find a basis for with the property that is diagonal.