linear algebra - When is matrix multiplication commutative ...
Multiplication of blocks will give diagonal $\lambda_1\lambda_2$, first off-diagonal $\lambda_1 + \lambda_2$ and second off-diagonal $1$ so assuming scalar multiplication and addition is commutative so will the jordan blocks.
How to write the derivative of a matrix product with respect to a …
To understand what "of total order $3$" means, consider this idea : if you have a vector, taking inner products with a vector of same length (a tensor of order $1$) gives you a number. If you have a matrix, taking the products with two vectors, one for each dimension of …
Matrix multiplication notation - Mathematics Stack Exchange
Matrix multiplication notation. Ask Question Asked 7 years, 10 months ago. Modified 3 years, 6 months ago. ...
linear algebra - Dot product vs Matrix multiplication, is the later a ...
Which if we write in matrix form, we need to mathematically take the transpose of a vector and do 'matrix' multiplication to get the above dot product. So coming back full circle to the question - matrix multiplication is a tool to find vector dot product (assuming we are talking about matrices in the context of vectors)
Why, historically, do we multiply matrices as we do?
The eventual coefficients obtained from a double transformation of (x, y, z) require row elements of the left transform matrices to be multiplied by their corresponding column element of the right transform matrix. My own old idea was that if multiplication of a vector by a matrix is done like A (x, y, z) = (x', y', z'), i.e. using row vectors ...
linear algebra - Proof of # of FLOPs in Matrix Multiplication ...
2020年1月17日 · Since you have p p products, you add p − 1 p − 1 of them to the first one. So the number of operations for one element in the output matrix is p p multiplications and p − 1 p − 1 additions, meaning 2p − 1 2 p − 1 FLOPS. Then for all elements, you get n × m × (2p − 1) n × m × (2 p − 1) FLOPS. I understand that is how it ...
associativity - Why is matrix multiplication associative?
2020年3月16日 · The proof below is a straightforward application of the definition matrix multiplication, and you can probably find it any linear algebra text. The only "trick" would be switching the order summation in a double summation. Although, if you have a lot of experience with sums, then you probably don't think of this as a "trick".
linear algebra - Practical uses of matrix multiplication
2011年5月28日 · Matrix multiplcation plays an important role in quantum mechanics, and all throughout physics. Examples include the moment of inertia tensor, continuous-time descriptions of the evolution of physical systems using Hamiltonians (especially in systems with a finite number of basis states), and the most general formulation of the Lorentz transformation from special …
Is there a 3-dimensional "matrix" by "matrix" product?
A matrix represents any finite-discrete linear transform of field values. A matrix is a finite-discrete collection of field values. So if you have a linear transform that converts one matrix to another matrix, then the transform itself can be represented with matrix multiplication.
block matrix multiplication - Mathematics Stack Exchange
If you partition after x rows in first matrix , you've to partition after x columns (not rows ) in the second matrix. Otherwise while multiplying you'll have to multiply mn block with another mn block which is not possible. (you need np block) Try it with your example. – A Googler. CommentedOct 1, 2015 at 18:08. 1.