Matrix Transpose & Symmetric Matrices | Linear Algebra #5

📚About
The fifth lecture of the "Linear Algebra" series is entitled "Matrix Transpose & Symmetric Matrices". This lecture is two-fold, focusing on Non-singularity and Linear Systems.
The transpose of an m × n matrix A, named A^T, is the n × m matrix obtained by exchanging the rows and columns of A. We also give the most important properties of matrix transposition.
An important class of matrices are the symmetric matrices. A square matrix A is symmetric if A^T = A, and this means that a_{ij} = a_{ji}, where 1 ≤ i, j ≤ n. Many problems in engineering and science involve symmetric matrices, and entire sections of this book deal with them. As you will see, when a problem involves a symmetric matrix, this normally leads to a faster and more accurate solution. It is of the utmost importance that you remember the relationship (AB)^T = B^TA^T, as we will use it again and again throughout this course (probably without mentioning that it is actually a property).

00:00 What is a Transpose ?
01:14 Transposition Properties
02:59 What is a Symmetric Matrix ?
04:12 Remarks on a special symmetric matrix (A^TA)
05:24 Summary

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[lawrenceharbin]

You rock Dr. BAZZI. Please stay as consistent as this looks !!

[kamrynhauck]

great initiative to start linear algebra. I so need this.

[MrPlanetone]

Yes you are right in 5:57, Indeed, many applications of symmetric 'm working with a project including matrix representation of undirected graphs and there it happens that the matrix representing the graph should be mean think about it if you go from node i to node j it's the same thing all around.

[zaraduncan]

This media has navigated into my heart💘

[jennhilger]

in 2:08, so that essentially means whatever you do to A applies to its transpose as well, right ?

[alethaprohaska]

So in 2:08 it’s as if whatever you could say about A, then it applies to its transpose as well, no?

[brucepeters]

Btw, a note on 5:57, the famous Minres method (based on tridiagonalization) in the domain of Kyrlov subspaces, does really really well when a symmetric matrix is utilized (given that it is well-conditioned).I enjoyed your linear algebra series, please go on ;) Good luck from the USA.

[SolvingOptimizationProblems]

May I ask a question Dr. Bazzi? How did you make this video? What software did you use? Thank you very much!

[joelmarshall]

great videos and I'm sure this is going to turn out to be a great live premiere !! Good job Dr. Ahmad Bazzi.

[leahprice]

3:41 why couldn’t it work for rectangular matrices?

[jerryhernandez]

5:50 yeah that’s what my professor says, if A=A^t then the matrix is symmetrical. Never understood it until now, yey..

[megcummings]

Thanks for the lecture, when is the next one coming out?

[SmijeexD]

amazing lecture, I just subscribed to your it up

[porterchamplin]

OK linear systems are nice but how can we solve them ? do you think you can conduct a lecture on solving them. I also see you will talk about cubic spline interpolations, that's interesting. Waiting for the stream :)

[cameronwoodward]

Please conduct a lesson on Gaussian elimination for solving linear systems. I know it’s related to invertible matrices and if there is all zero rows means the matrix is singular. Could you please speak on that matter? Thanks.

[miawalker]

Please lecture on Gaussian elimination,

[jordonwindler]

Do you think you can do a numerical practical example on trusses, I'm a civil engineer and would love to see how linear systems apply to real life trusses.

[marthaporter]

When is the lecture on Homogenous systems ?

[pisellina]

The easiest thing in linear algebra is when you multiply a scalar with a matrix as in could absorb it in or throw it anywhere outside.

[michaelbeamon]

ok how does all this relate to rank(A), which stands for rank of a matrix. If it doesn't, then when is the matrix rank lecture ?