Linear Algebra, 2020a

Section II. Similarity 407
̌ 1.13 [Halmos] Are there two matrices A and B that are similar while A 2 and B 2 are
not similar?
̌ 1.14 Prove that if two matrices are similar and one is invertible then so is the other.
1.15 Show that similarity is an equivalence relation. (The definition given earlier
already reflects this, so instead start here with the definition that ˆT is similar to T
if ˆT = PTP −1 .)
1.16 Consider a matrix representing, with respect to some B, B, reflection across
the x-axis in R 2 . Consider also a matrix representing, with respect to some D, D,
reflection across the y-axis. Must they be similar?
1.17 Prove that matrix similarity preserves rank and determinant. Does the converse
hold?
1.18 Is there a matrix equivalence class with only one matrix similarity class inside?
One with infinitely many similarity classes?
1.19 Can two different diagonal matrices be in the same similarity class?
̌ 1.20 Prove that if two matrices are similar then their k-th powers are similar when
k>0. What if k 0?
̌ 1.21 Let p(x) be the polynomial c n x n + ···+ c 1 x + c 0 . Show that if T is similar to
S then p(T) =c n T n + ···+ c 1 T + c 0 I is similar to p(S) =c n S n + ···+ c 1 S + c 0 I.
1.22 List all of the matrix equivalence classes of 1×1 matrices. Also list the similarity
classes, and describe which similarity classes are contained inside of each matrix
equivalence class.
1.23 Does similarity preserve sums?
1.24 Show that if T − λI and N are similar matrices then T and N + λI are also
similar.
II.2
Diagonalizability
The prior subsection shows that although similar matrices are necessarily matrix
equivalent, the converse does not hold. Some matrix equivalence classes break
into two or more similarity classes; for instance, the nonsingular 2×2 matrices
form one matrix equivalence class but more than one similarity class.
The diagram below illustrates. Solid curves show the matrix equivalence
classes while dashed dividers mark the similarity classes. Each star is a matrix
representing its similarity class. We cannot use the canonical form for matrix
equivalence, a block partial-identity matrix, as a canonical form for similarity
because each matrix equivalence class has only one partial identity matrix.
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