- #1

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For example, does this always hold true?

M_ab = v_a × w_b

If not, where does it break down?

M_ab = v_a × w_b

If not, where does it break down?

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- #1

- 236

- 40

For example, does this always hold true?

M_ab = v_a × w_b

If not, where does it break down?

M_ab = v_a × w_b

If not, where does it break down?

- #2

Mentor

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https://en.wikipedia.org/wiki/Outer_product

https://stattrek.com/matrix-algebra/vector-multiplication.aspx

- #3

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##AB = \begin{bmatrix}ac & ad \\ bc & bd\end{bmatrix} = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}##.

Is this possible, knowing that ##xy = 0## for real numbers ##x,y## implies that either ##x=0## or ##y=0## ?

- #4

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##AB = \begin{bmatrix}ac & ad \\ bc & bd\end{bmatrix} = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}##.

Is this possible, knowing that ##xy = 0## for real numbers ##x,y## implies that either ##x=0## or ##y=0## ?

Thanks. I was trying to think of a counter example. This is very obvious.

- #5

Science Advisor

Homework Helper

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No, such a matrix has rank 1. Using the notation that ##\mathbf w^T## is the row vector of ##\mathbf w##, then row 1 of *M* is ##v_1 \mathbf w^T##, row 2 is ##v_2 \mathbf w^T##, etc. Every row is a multiple of every other row. And every column is a multiple of every other column, as all have the form ##w_j \mathbf v##.

And in particular, any invertible matrix 2x2 or larger is going to be a counterexample.

What is true is that you can express any matrix*M* of rank *n* as a sum of *n* rank-1 matrices ##\sum_{i=1}^n {\mathbf v_i \mathbf w_i^T}##.

And in particular, any invertible matrix 2x2 or larger is going to be a counterexample.

What is true is that you can express any matrix

Last edited:

- #6

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It can always be done by a linear combination of those where ##\operatorname{rank}M## is the minimal length of it.For example, does this always hold true?

M_ab = v_a × w_b

If not, where does it break down?

- #7

Science Advisor

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For example, does this always hold true?

M_ab = v_a × w_b

If not, where does it break down?

You didn't say how you define the "product of two vectors". Let's assume you are thinking of this type of example.

v_a = (a1, a2)

v_b = (b1,b,2,b3)

The "product" is a table of data with 2 rows and 3 columns. The (i,j) entry of the table is (a_i)(b_j).

It would be nice if all data tables were so simple! A person who could do multiplication wouldn't need the body of the table.

Thinking about the various complicated data tables that we encounter makes it clear that not all data tables (matrices) have such a simple structure.

However, as @fresh_42 indicates in post #6, all data tables can be written as

- #8

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- #9

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Sure, they are rank one matrices.

The question gets interesting, if we ask for the minimal length of linear combinations of ##x\otimes y \otimes z## to represent a given bilinear mapping, e.g. matrix multiplication. If we define the

Two is the lower and three the trivial upper bound. I find this fascinating, as it somehow contains the question whether there are intrinsically difficult problems out there, or whether we just haven't found the right clue - similar to NP=P and ERH.

- #10

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The question gets interesting, if we ask for the minimal length of linear combinations of ##x\otimes y \otimes z## to represent a given bilinear mapping, e.g. matrix multiplication. If we define thematrix exponent## \omega := \min\{\,\gamma\,|\,(A;B) \longmapsto A\cdot B = \sum_{i}^Rx_i(A) \otimes y_i(B) \otimes Z_i \,\wedge \, R=O(n^\gamma)\,\} ## then ##2\leq \omega \leq 2.3727## and we do not know how close we can come to the lower bound.

Thanks, this was really an interesting topic based on a thesis I found with Google search, and doesn't require too many concepts unfamiliar for me to understand.

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