Equivalence of Matrices

Let $A$ and $B$ be two matrices of order $m\times n$.Then:

<aside> 💡 $A$ is equivalent to $B$ if $B=QAP$ for some invertible $n\times n$ matrix$~P$ and for some invertible $m\times m$ matrix $Q$.

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Other Characteristics:

1. $A$ can be transformed to $B$ by a combination of elementary row and column operations.
2. $rank(A)=rank(B)$

• Equivalence of matrices is an equivalence relation:
  1. $A$ is equivalent to itself. | $A=I_{m\times m} A I_{n\times n}$
  2. $A$ is equivalent to $B$ implies $B$ is equivalent to $A$. | $B=QAP \implies A =Q^{-1}BP^{-1}$
  3. $A$ is equivalent to $B$ and $B$ to $C$ implies $A$ is equivalent to $C.$ | $B=QAP ~\& C=Q^{'}BP^{'} \implies C=Q^{'}(QAP)P^{'} \implies C=Q^{''}AP^{''}$

Linear Transformations and Equivalence of Matrices.

Consider a linear transformation $T:V\rightarrow W$, two ordered bases $\beta_{1}$ and $\beta_2$ for $V$, and two ordered bases $\gamma_1$ and $\gamma_2$ for $W$.

Let $A$ be the matrix corresponding to $T$ with respect to the bases $\beta_1$ and $\gamma_1$ and $B$ be the matrix corresponding to $T$ with respect to the bases $\beta_2$ and $\gamma_2$.

<aside> 💡 Then $A$ is equivalent to B.

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For $P$ → express the ordered basis $\beta_2$ in terms of ordered basis $\beta_1$. For $Q$ → express the ordered basis $\gamma_1$ in terms of ordered basis $\gamma_2$.

Similar Matrices

<aside> 💡 An $n\times n$ matrix $A$ is similar to an $n\times n$ matrix $B$ if there exists an $n\times n$ invertible matrix $P$ such that $B=P^{-1}AP$.

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Similarity is an equivalence relation:

1. $A$ is similar to itself.
2. $A$ is similar to $B$ implies $B$ is similar to $A$.
3. $A$ is similar to $B$ and $B$ to $C$ implies $A$ is similar to $C$.

Important Properties of Similar Matrices.

Suppose $A$ and $B$ are similar matrices. Then the following properties hold:

<aside> 💡 1. $A$ and $B$ are equivalent. 2. $A$ and $B$ have the same rank. 3. $det(B)=det(P^{-1}AP)=det(P^{-1})det(A)det(P)=det(A)$

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