Similar Matrices and Similarity Invariants | Linear Algebra

We introduce similar matrices, similarity transformations, similarity invariants, we prove some similarity invariants, prove transitivity of similar matrices, go over how to show 2 matrices are similar given an invertible matrix P that creates the similarity transformation, and do three examples showing matrices aren't similar. #linearalgebra
Proving Similarity Invariants: (coming soon)
Diagonalizing a Matrix: • Diagonalizing Matrices...
Linear Algebra course: • Linear Algebra
Linear Algebra exercises: • Linear Algebra Exercises
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0:00 Intro
0:19 Similarity Transformations
0:49 Similarity Invariants
2:14 Proof Similar Matrices have Same Determinant
3:16 Proof Similar Matrices Share Invertibility
4:42 Definition of Similar Matrices
5:21 Symmetry of Similar Matrices
6:24 Transitivity of Similar Matrices
7:38 Showing Two Matrices are Similar
9:41 Showing Two Matrices aren't Similar
11:31 Conclusion
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[@talastra]

phew ... what a density of information.

[@WrathofMath]

Yeah, covered a lot in this one!

[@adamjahani4494]

Can you also use the formula BP=AP to show similarity for the 2 by 2 matrix

[@WrathofMath]

No, the matrix P has to be invertible, in which case your equation would imply B=A. Perhaps you had a typo; we could have PB = AP with an invertible matrix P to show similarity.

[@user-wr4yl7tx3w]

When you say eigenspace are you referring to the eigenvectors?

[@WrathofMath]

When I refer to an eigenspace corresponding to an eigenvalue, I am referring to the subspace of R^n consisting of the eigenvectors corresponding to the eigenvalue and also including the zero vector

[@talastra]

@@WrathofMath Also known as, see other video!

[@raulbeienheimer]

If I don't recall it wrongly, similarity can be viewed as an isomorphism?

[@WrathofMath]

Thanks for watching; and indeed! For example it would be an isomorphism on the multiplicative group of invertible nxn matrices.