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In: Math

"If A and B are similar then A and B are orthogonally similar. " Prove or...

"If A and B are similar then A and B are orthogonally similar. " Prove or disprove this statement.

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Expert Solution

Is it necessary true that,

  1. Every two Invertible matrices with the same dimension are similar to each other.
  2. Every two row equivalence matrices with the same dimension are similar to each other.

If one of them is wrong, give an example. Otherwise, prove.

Well for the first one it is necessary true that every 2 orthogonal matrices with the same dimension are similar to each other because after doing elementary row operations on both of them we get the same Identity matrix (because the dimension is the same). And then we can say that if M is orthogonal then and A,B are those two matrices:

M?1AM=B?M?1IM=I?I=Iwhich is true.

For the second one, well I didn't give an example, but I'm pretty sure that if we take two matrices with the same dimension with the same row equivalence they won't necessary be similar to EVERY TWO matrices that we take because:

Assume A?In and A is orthogonal matrix then: (also: A is row equivalent to In)

A=M?1InM?A=M?1M?A=InSo A=In which can't be true because A?I so A can be row equivalent but can't be similar.

milcah answered 1 year ago
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