Falk's scheme

The Falk schema (named after the German engineer Sigurd Falk ) is a table that provides visual aid for the multiplication of matrices by hand. The left factor that - the matrix is left of the -Ergebnismatrix and the right factor, the matrix is placed above the result matrix. The corresponding scalar product is entered where the -th row of the left multiplicand and the -th column of the right multiplicand cross . ${\displaystyle (m\times r)}$ ${\displaystyle (m\times n)}$ ${\displaystyle (r\times n)}$ ${\displaystyle i}$ ${\displaystyle j}$

example

The matrices are given

${\displaystyle A_{3\times 2}={\begin{pmatrix}1&4\\2&5\\3&-6\end{pmatrix}}}$ and . ${\displaystyle B_{2\times 2}={\begin{pmatrix}-1&1\\1&-2\\\end{pmatrix}}}$

The product is to be determined. is a matrix. ${\displaystyle C=A\cdot B}$ ${\displaystyle C}$ ${\displaystyle (3\times 2)}$

First, the Falk schema is set up by writing the matrices next to one another with a height offset (in the original orientation, i.e. without tilting or rotating).

			Column j
			1	2
			−1	1
Line i			1	−2
1	1	4th
2	2	5
3	3	−6

The first row of is multiplied element by element by the first column of : 1 · (−1) + 4 · 1 = 3 and results in the element . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle c_{11}=3}$

			Column j
			1	2
			−1	1
Line i			1	−2
1	1	4th	3
2	2	5
3	3	−6

The first row of is multiplied element by element by the second column of : 1 · 1 + 4 · (−2) = −7 and results in the element . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle c_{12}=-7}$

			Column j
			1	2
			−1	1
Line i			1	−2
1	1	4th	3	−7
2	2	5
3	3	−6

The same procedure is used for the other lines. Finally, the third row of is multiplied by the second column of : 3 · 1 + (−6) · (−2) = 15 and results in the element . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle c_{32}=15}$

			Column j
			1	2
			−1	1
Line i			1	−2
1	1	4th	3	−7
2	2	5	3	−8
3	3	−6	−9	15th

literature

• Sigurd Falk: A clear scheme for matrix multiplication . In: Journal for Applied Mathematics and Mechanics (ZAMM) . tape 31 , no. 4-5 , 1951, ISSN  0044-2267 , pp. 152–153 , doi : 10.1002 / zamm.19510310409 .
• Rudolf Zurmühl , Falk: Matrices and their application , Volume 1, Springer, 7th edition 1997, p. 17
• Sascha Kurz, Jörg Rambau: Mathematical foundations for economists . Kohlhammer Verlag, Stuttgart 2009, ISBN 978-3-17-019882-1 , pp. 29-30 .
• Lothar Papula : Mathematics for Engineers and Natural Scientists . 4th edition. tape 2 . Vieweg + Teubner Verlag, Wiesbaden 2010, ISBN 978-3-8348-9730-5 , pp. 525-528 .
• Karl-Eugen Kurrer : The History of the Theory of Structures. Searching for Equilibrium , Berlin: Ernst & Sohn 2018, pp. 842f., ISBN 978-3-433-03229-9

Web links

• Wikibooks: Analytical Geometry - Matrices - Computing with Matrices - Matrix Multiplication
• Dankert (HAW Hamburg): Various examples and their expansion