Suppose that A and B are diagonalizable matrices. Prove or disprove
that A is similar to B if and only if A and B are unitarily equivalent.

Efan Halliday
2021-01-02
Answered

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SchulzD

Answered 2021-01-03
Author has **83** answers

Step 1

Consider A and B are matrices that are diagonalizable.

The objective is to prove or disprove that A is similar to B only if A and B are unitarily equivalent.

Consider the two diagonalizable matrices are,

$A=\left[\begin{array}{cc}1& -1\\ 0& 0\end{array}\right]$

$B=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$

Consider that there is an invertible matrix,

$P=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$
Step 2

The matrix A is similar to B as shown below,

$\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]={\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]}^{-1}\left[\begin{array}{cc}1& -1\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$

$=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$

$=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$

The A and B are similar matrices.

But A and B are not unitary because B is symmetric, but A is not.

Also, recall the result, If B and A are unitarily equivalent.

$\sum _{i,j=1}^{n}|{b}_{ij}{|}^{2}=\sum _{i,j=1}^{n}|{a}_{ij}{|}^{2}$

Since, 2 is not equal then,

$\sum _{i,j=1}^{n}|{b}_{ij}{|}^{2}\ne \sum _{i,j=1}^{n}|{a}_{ij}{|}^{2}$

Since,

A and B are not unitarily equivalent.

Therefore, it is not necessary that A is similar to B if and only if A and B are unitarily equivalent.

Consider A and B are matrices that are diagonalizable.

The objective is to prove or disprove that A is similar to B only if A and B are unitarily equivalent.

Consider the two diagonalizable matrices are,

Consider that there is an invertible matrix,

The matrix A is similar to B as shown below,

The A and B are similar matrices.

But A and B are not unitary because B is symmetric, but A is not.

Also, recall the result, If B and A are unitarily equivalent.

Since, 2 is not equal then,

Since,

A and B are not unitarily equivalent.

Therefore, it is not necessary that A is similar to B if and only if A and B are unitarily equivalent.

Jeffrey Jordon

Answered 2022-01-23
Author has **2027** answers

Answer is given below (on video)

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3. add row 3 to row 1,

4. interchange columns 1 and 4,

5. subtract row 2 from each of the other rows,

6. replace column 4 by column 3,

7. delete column 1 (column dimension is reduced by 1).

(a) Write the result as a product of eight matrices.

(b) Write it again as a product of ABC (same B) of three matrices.

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4. interchange columns 1 and 4,

5. subtract row 2 from each of the other rows,

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7. delete column 1 (column dimension is reduced by 1).

(a) Write the result as a product of eight matrices.

(b) Write it again as a product of ABC (same B) of three matrices.

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