A nice, concise definition of the trace, which makes it look a little less arbitrary-looking (if a little more technical) than the standard definition "the sum of elements on the diagonal", and also makes it more evident that it doesn't depend on the choice of coordinates:
The trace is the only linear mapping¹ such that:
trace(|u><v|) = <u|v>
for any vectors |u>, |v>.
(The <|> things are bracket notation: |u> stands for a regular vector (think "column vector"), <u|, for its corresponding dual vector (think "row vector"), |u><v| for the tensor product of |u> and |v>, and <u|v> to their dot product)
This still has the drawback of looking like it depends on a choice of inner product, when it actually doesn't.
¹ From linear mappings from V to V, to the scale field K.
Grégoire Locqueville
npub1mg…94dr4
2025-12-19 20:47:01 UTC
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npub1mghlhx2z6vnmvhhxdu2j66dyylc3ff9y9a56fkdc32uu4wtr8xust94dr4Seen on
wss://relay.ditto.pubPublished at
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"content": "A nice, concise definition of the trace, which makes it look a little less arbitrary-looking (if a little more technical) than the standard definition \"the sum of elements on the diagonal\", and also makes it more evident that it doesn't depend on the choice of coordinates:\n\nThe trace is the only linear mapping¹ such that:\ntrace(|u\u003e\u003cv|) = \u003cu|v\u003e\nfor any vectors |u\u003e, |v\u003e.\n\n(The \u003c|\u003e things are bracket notation: |u\u003e stands for a regular vector (think \"column vector\"), \u003cu|, for its corresponding dual vector (think \"row vector\"), |u\u003e\u003cv| for the tensor product of |u\u003e and |v\u003e, and \u003cu|v\u003e to their dot product) \n\nThis still has the drawback of looking like it depends on a choice of inner product, when it actually doesn't.\n\n¹ From linear mappings from V to V, to the scale field K.",
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