Linear Algebra/Definition and Examples of Similarity

Definition and Examples

We've defined $H$ and $\hat{H}$ to be matrix-equivalent if there are nonsingular matrices $P$ and $Q$ such that $\hat{H}=PHQ$. That definition is motivated by this diagram

showing that $H$ and $\hat{H}$ both represent $h$ but with respect to different pairs of bases. Template:AnchorWe now specialize that setup to the case where the codomain equals the domain, and where the codomain's basis equals the domain's basis.

To move from the lower left to the lower right we can either go straight over, or up, over, and then down. In matrix terms,

${\mathrm{R}\mathrm{e}\mathrm{p}}_{D,D}(t)={\mathrm{R}\mathrm{e}\mathrm{p}}_{B,D}(\text{id}){\mathrm{R}\mathrm{e}\mathrm{p}}_{B,B}(t)({\mathrm{R}\mathrm{e}\mathrm{p}}_{B,D}(\text{id}){)}^{-1}$

(recall that a representation of composition like this one reads right to left).

Since nonsingular matrices are square, the similar matrices $T$ and $S$ must be square and of the same size.

Since matrix similarity is a special case of matrix equivalence, if two matrices are similar then they are equivalent. What about the converse: must matrix equivalent square matrices be similar? The answer is no. The prior example shows that the similarity classes are different from the matrix equivalence classes, because the matrix equivalence class of the identity consists of all nonsingular matrices of that size. Thus, for instance, these two are matrix equivalent but not similar.

$T=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)S=\left(\begin{array}{cc}1& 2\\ 0& 3\end{array}\right)$

So some matrix equivalence classes split into two or more similarity classes— similarity gives a finer partition than does equivalence. This picture shows some matrix equivalence classes subdivided into similarity classes.

To understand the similarity relation we shall study the similarity classes. We approach this question in the same way that we've studied both the row equivalence and matrix equivalence relations, by finding a canonical form for representatives of the similarity classes, called Jordan form. With this canonical form, we can decide if two matrices are similar by checking whether they reduce to the same representative. We've also seen with both row equivalence and matrix equivalence that a canonical form gives us insight into the ways in which members of the same class are alike (e.g., two identically-sized matrices are matrix equivalent if and only if they have the same rank).

Exercises

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References

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