Learning about tensor transformation (See my tensor page in the learning math menu option) I came into Jakobian Matrix calculation. I studied matrices at Chalmers in Göteborg as a tool to solve equations.
I need to learn Matrix calculus.
Khan Academy has a great introduction to matrices in www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices/x9e81a4f98389efdf:mat-intro/a/intro-to-matrices
Matrix calculus is what I need most to be able to continue with tensor calculations.
Number of columns in the first matrix must be equal to the number of raws in the second matrix.
If it is not, the multiplication can not be done = it is not defined.
The number of columns in the second matrix decides the number of columns in the resulting/ matrix product.
- A23 x B32 gives AB22
This multiplication can be commuted to
- B32 x A23x gives AB23
(Again, as said above the number of columns in A decides the number of columns in AB)
You can not multiply A12 x B32 as the number of raws in B is not equal to the number of columns in A.
This is for me, the most important part.
- E23xD32 =ED22
Multiplying matrix with identity matrix (I)
If you multiply a matrix A with a identity matrix of A you get A
I x A = A x I = A
OBS. Multiplication with identity matrix is commutative if you have square matrices.
How to calculate A-1 the inverted Matrix of A
If A x A-1 = I (he identity matrix) then you have found the inverted matrix of A.
To find the inverted matrix of a square 2×2 matrix you have to use this formula
Khan explains in a to difficult way how to invert a square matrix so I choosed this video instead from Professor Dave Explains
This is Khans explanation, for you to compare.
When matrix multiplication are commutative like a x b = b x a
- D33B22 = not defined! (numbers of rows not equal to numbers of columns in B)
- Matrix multiplication is therefore not commutative in this case
- A52B23= not defined (numbers of rows not equal to numbers of columns in B)