Linear Algebra Lecture 9.1 Normal Operators

00:00 Recap
00:20 Problems
02:37 Normal Operators
06:15 An Important Lemma
09:32 A Criterion for Normality
12:46 A Corollary
13:06 Normal Operators and Invariant Subspaces

Note: The auto-generated subtitles are mostly accurate and may enhance experience.

Prerequisites: Familiarity with mathematical notation and terminology.

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[problem]

You can think of it in this way, every matrix with linearly independent eigen vectors can be decomposed into (U)(D)(U^-1), now suppose M is some matrix whose eigen vectors are orthogonal so M=(P)(D)(P^T) we can immediately say if we multiply from left or right by transpose of M i.e M^T we will get same result (M)(M^T)=(M^T)(M) because transpose of transposed P will be P, so every matrix which have orthogonal eigen vectors will satisfy this property, also if some matrix M=(P)(D)(P^T) so if we take the transpose of both sides we get M^T=(P)(D^T)(P^T) but if M have real eigen values so the conjugate transpose of D will be equal to D as it is only diagonal matrix so we will have M^T=(P)(D)(P^T), so every matrix which will satisfy this property that M=M^T will also have real eigen values, also orthogonal eigen vectors, (transpose also means conjugate transpose and diagonal matrix D by theorem has eigen values of M on its diagonal)