Trace (mathematics) | Wikipedia audio article
Channel: wikipedia tts
Date: 07/10/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Trace_(linear_algebra)


00:00:58 1 Definition
00:01:57 2 Example
00:02:55 3 Properties
00:03:25 3.1 Basic properties
00:04:23 3.2 Trace of a product
00:05:22 3.3 Cyclic property
00:06:20 3.4 Trace of a matrix product
00:06:50 3.5 Trace of a Kronecker product
00:07:48 3.6 Full characterization of the trace
00:08:47 3.7 Similarity invariance
00:09:46 3.8 Trace of product of symmetric and skew-symmetric matrix
00:10:44 3.9 Relation to eigenvalues
00:11:43 3.9.1 Trace of the identity matrix
00:12:12 3.9.2 Trace of an idempotent matrix
00:13:11 3.9.3 Trace of a nilpotent matrix
00:13:40 3.9.4 Trace equals sum of eigenvalues
00:14:39 3.10 Trace of commutator
00:15:08 3.11 Trace of powers of nilpotent matrices
00:16:07 3.12 Trace of Hermitian matrix
00:17:05 3.13 Trace of projection matrix
00:17:34 4 Exponential trace
00:18:33 5 Trace of a linear operator
00:19:32 5.1 Eigenvalue relationships
00:20:30 5.2 Derivatives
00:21:00 6 Applications
00:21:58 7 Lie algebra
00:22:57 7.1 Bilinear forms
00:23:55 8 Inner product
00:24:54 9 Generalizations
00:25:53 10 Coordinate-free definition
00:27:21 10.1 Dual
00:28:19 11 See also
00:29:47 12 Notes
00:31:15 13 References
00:32:43 14 External links
00:34:11 Coordinate-free definition
00:36:37 j and 0 otherwise. This shows that t(A) is simply the sum of the coefficients along the diagonal. This method, however, makes coordinate invariance an immediate consequence of the definition.
00:37:36 Dual
00:38:35 See also



Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.

Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain

Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.

Listen on Google Assistant through Extra Audio:
https://assistant.google.com/services/invoke/uid/0000001a130b3f91
Other Wikipedia audio articles at:
https://www.youtube.com/results?search_query=wikipedia+tts
Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
Speaking Rate: 0.7143054516724755
Voice name: en-GB-Wavenet-B


"I cannot teach anybody anything, I can only make them think."
- Socrates


SUMMARY
=======
In linear algebra, the trace (often abbreviated to tr) of a square matrix A is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of A.
The "trace" of a matrix is sum of the (complex) eigenvalues, and it is invariant with respect to a change of basis. This characterization can be used to define the "trace" of a linear operator in general. The "trace" is only defined for a square matrix (n × n).
The trace is related to the derivative of the determinant (see Jacobi's formula).