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Mathematical Expression Editor
Similar Matrices and Their Properties
Let and be matrices. Then the products and are both matrices. In most cases the
products and are not equal.
However, for some pairs of matrices and , we are able to find an invertible matrix
such that . This leads to the following definition.
If and are matrices, we say that and are similar, if for some invertible matrix .
In this case we write .
The following theorem shows that similarity () satisfies reflexive, symmetric, and
transitive properties.
Similarity is an equivalence relation, i.e. for matrices and ,
item:symmetric If then for some invertible matrix , \begin{equation*} A=P^{-1}BP \end{equation*}
and so \begin{equation*} PAP^{-1}=B \end{equation*}
But then \begin{equation*} \left ( P^{-1}\right ) ^{-1}AP^{-1}=B \end{equation*}
which shows that .
item:transitive Now suppose and . Then there exist invertible matrices such that \begin{equation*} A=P^{-1}BP,\ B=Q^{-1}CQ \end{equation*}
Then, \begin{equation*} A=P^{-1} \left ( Q^{-1}CQ \right )P=\left ( QP\right ) ^{-1}C\left ( QP\right ) \end{equation*}
showing that is similar to .
Any relation satisfying the reflexive, symmetric and transitive properties is called an
equivalence relation. Theorem th:similarityequivalence proves that similarity between matrices is an
equivalence relation. Practice Problem prob:lessthan gives a good example of a relation that is
NOT an equivalence relation.
As we will see later, similar matrices share many properties. Before proceeding to
explore these properties, we pause to introduce a simple matrix function that we will
continue to use throughout the course.
The trace of an matrix , abbreviated , is defined to be the sum of the main diagonal
elements of . In other words, if , then
We may also write .
It is easy to see that and that holds for all matrices and and all scalars . The
following fact is more surprising.
Let and be matrices. Then .
Proof
Write and . For each , the -entry of the matrix is given as follows: . Hence
\begin{equation*} \mbox{tr}(AB) = d_1 + d_2 + \dots + d_n = \sum _{i}d_i = \sum _{i}\left (\sum _{j}a_{ij}b_{ji}\right ) \end{equation*}
Similarly we have . Since these two double sums are the same, we have proven the
theorem.
The following theorem lists a number of properties shared by similar matrices.
Similarly, for th:properties_similar_rank , because multiplication by an invertible matrix cannot change
the rank. To see this, note that any invertible matrix is a product of elementary
matrices. Multiplying by elementary matrices is equivalent to performing
elementary row (column) operations on , which does not change the row
(column) space, nor the rank. It follows that similar matrices have the same
rank.
For th:properties_similar_trace, we make use of Theorem th:trAB=trBA: \begin{equation*} \mbox{tr} (P^{-1}AP) = \mbox{tr}[P^{-1}(AP)] = \mbox{tr}[(AP)P^{-1}] = \mbox{tr} A. \end{equation*}
As for th:properties_similar_char_poly, \begin{align*} \det (B-\lambda I) &= \det \{P^{-1}AP-\lambda (P^{-1}P)\} \\ &= \det \{ P^{-1}(A-\lambda I)P\} \\ &= \det (A-\lambda I), \end{align*}
Sharing the five properties in Theorem th:properties_similar does not guarantee that two matrices are
similar. The matrices and have the same determinant, rank, trace, characteristic
polynomial, and eigenvalues, but they are not similar because for any invertible
matrix .
Even though the properties in Theorem th:properties_similar cannot be used to show two matrices are
similar, these properties come in handy for showing that two matrices are NOT
similar.
Are the matrices and similar?
A quick check shows us , and both matrices are seen
to be invertible, so they have the same rank. However, and , so the matrices are not
similar.
The next theorem shows that similarity is preserved under inverses, transposes, and
powers:
At the beginning of this section we mentioned that similarity of matrices is an
equivalence relation.
An equivalence relation is a binary relation on elements of a set that has the
following properties:
The reflexive property: for every
The symmetric property: If , then for every
The transitive property: If and , then for every
Let be the set of all positive integers. We can show that the relation “less than”
(symbolized by ) is NOT an equivalence relation on this set. To see this, note
that “less than” is not reflexive, because is not true for any positive integer
.
(a)
Is the relation “less than” symmetric? YesNo
(b)
Is the relation “less than” transitive? YesNo
Another relation between matrices we have studied in this course is that two
matrices can be “row equivalent”. Is the relation “row equivalent”