f(“# u)= A”# u.

B = {“#!1,”#! 2}

f的特征向量，从标准基写下转移矩阵P
B0 到 B 并计算其逆 P！ 1. 通过直接计算检查

• 计算A 的特征多项式。
• 找出A 的特征值。
• 对于A 的每个特征值，找到相应特征空间的基。
• 通过将这些基放在一起，确定是否存在基
B = {“#! 1,
“# ! 2,…,
“#！n}
A的特征向量。确定A是否可对角化。

Exercise 1. For each one of the following 2 ! 2 matrices A, compute the
eigenvalues and the eigenspaces of A and draw the eigenspaces of A in the xy
plane. Describe the action of the linear transformation f from R2 into itself
deﬁned, for each “# u \$ R2,by

f(“# u)= A”# u.

Is the map f diagonalizable? If it is, ﬁnd a basis
B = {“# ! 1,”# ! 2}

of eigenvectors of f, write down the transition matrix P from the standard basis
B0 to B and compute its inverse P!1. Check by direct computation that
B = P!1AP

is a diagonal matrix.

Exercise 2. We want to diagonalize if possible each one of the following n!n
matrices A. We proceed in the following steps:
• Compute the characteristic polynomial of A.
• Find the eigenvalues of A.
• For each eigenvalue of A, ﬁnd a basis of the corresponding eigenspace.
• By putting these bases together, determine if there exists a basis
B = {“# ! 1,
“# ! 2,…,
“# ! n}
of eigenvectors of A.Determineif A is diagonalizable or not.

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