MATRIX DEFINITION A matrix is any doubly subscripted array of elements arranged in rows and columns.

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1 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review MTRIX DEFINITION mtrix is ny douly suscripted rry of elements rrnged in rows nd columns. m - Column Revised /0 n -Row m,,,,,, n n mn ij nd Order Tensor Slide of

2 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review MTRIX DEFINITION DIFFERENT NOTTIONS ij ll the Sme Vector! Revised /0 Slide of

3 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review MTRIX DEFINITION ROW VECTOR [ x n] Mtrix,,, n j j Indicil Nottion for Rows Revised /0 Slide of

4 [m x ] Mtrix 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review MTRIX DEFINITION COLUMN VECTOR m i i Indicil Nottion for Columns Revised /0 Slide 4 of

5 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review MTRIX DEFINITION SQURE MTRIX [m x n] Mtrix, where m = n B Revised /0 Slide 5 of

6 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review MTRIX DEFINITION DETERMINNT OF SQURE MTRIX = det[] = Determinnt of c d d c Revised /0 Slide 6 of

7 Revised /0 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review Slide 7 of h i g f e d c B MTRIX DEFINITION DETERMINNT OF SQURE MTRIX B = det[b] = Determinnt of B h g e d c i g f d i h f e B

8 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review MTRIX DEFINITION DETERMINNT OF SQURE MTRIX Summry For mtrix the determinnt is d - c For mtrix multiply y the determinnt of the mtrix tht is not in 's row or column, likewise for nd c, ut rememer tht hs negtive sign! The pttern continues for lrger mtrices: multiply y the determinnt of the mtrix tht is not in 's row or column, continue like this cross the whole row, ut rememer the pttern. Revised /0 Slide 8 of

9 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review MTRIX DEFINITION IDENTITY MTRIX Squre mtrix with ones on the digonl nd zeros elsewhere. I = [ ij ] = 0 if i j I = if i = j Revised /0 Slide 9 of

10 Revised /0 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review Slide 0 of ji mn n m m,,,,,, T Rows ecome columns nd columns ecome rows MTRIX DEFINITION TRNSPOSE MTRIX ij mn m n n,,,,,,

11 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review MTRIX DDITION new mtrix C my e defined s the dditive comintion of mtrices nd B where: C = + B is defined y: C B ij Note: ll three mtrices re of the sme dimension ij ij Revised /0 Slide of

12 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review MTRIX DDITION If nd B then B C ( ( ) ) ( ( ) ) Revised /0 Slide of

13 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review MTRIX DDITION EXMPLE B C Revised /0 Slide of

14 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review MTRIX SUBTRCTION new mtrix C my e defined s the sutrctive comintion of mtrices nd B where: C = - B is defined y: C B ij Note: ll three mtrices re of the sme dimension ij ij Revised /0 Slide 4 of

15 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review MTRIX SUBTRCTION If nd B then B C ( ( ) ) ( ( ) ) Revised /0 Slide 5 of

16 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review MTRIX SUBTRCTION EXMPLE B C Revised /0 Slide 6 of

17 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review MTRIX MULTIPLICTION Mtrices nd B hve these dimensions: [r x c] nd [s x d] Mtrices nd B cn e multiplied if: [r x c] nd [s x d] c = s Revised /0 Slide 7 of

18 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review MTRIX MULTIPLICTION The resulting mtrix C will hve dimensions of: [r x c] nd [s x d] C = r x d Revised /0 Slide 8 of

19 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review MTRIX MULTIPLICTION EXMPLE: C = + B [ x ] Mtrix [ x ] Mtrix B C Revised /0 [ x ] Mtrix Slide 9 of

20 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review MTRIX MULTIPLICTION EXMPLE: C = + B 0 B 0 [ x ] [ x ] nd B cn e multiplied Revised /0 C * * 5 ** * 0* * *0 **0 * 0*0 * * 8 ** * 0* 5 8 [ x ] Slide 0 of

21 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review MTRIX INVERSION B B BB I Like reciprocl in sclr mth Like the numer one in sclr mth Revised /0 Slide of

22 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review LINER SYSTEM OF SIMULTNEOUS EQUTION Two () Simultneous Equtions x + x = x + x = I X B X x x B Equtions in Mtrix Form Multiply y Inverse of X B X I X X B B B Revised /0 Slide of

23 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review LINER SYSTEM OF SIMULTNEOUS EQUTIONS: EXMPLE First precinct: 6 rrests lst week eqully divided etween felonies nd misdemenors. Second precinct: 9 rrests - there were twice s mny felonies s the first precinct. st Precinct : x x 6 nd Pr ecinct : x x 9 Revised /0 Slide of

24 Revised /0 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review Slide 4 of 9 6 * x x x x ; 9 6 * * * x x Premultiply oth sides y inverse mtrix * 0 0 x x squre mtrix multiplied y its inverse results in the identity mtrix. [x] identity mtrix multiplied y the [x] mtrix results in the originl [x] mtrix. LINER SYSTEM OF SIMULTNEOUS EQUTIONS: EXMPLE nswer

25 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review DDITIONL MTRIX OPERTIONS ssocitive Rule Commuttive Rule B C B C B B B C BC B C B C T T B B - - B B T - - T T - -T Revised /0 Slide 5 of

26 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review Nottion: V i V ij V ijk V ijkl TENSORS = memers = 9 memers = 7 memers 4 = 8 memers Kronecker s Delt ij = 0 if i j = if i = j ii = + + ii = + + ii = Revised /0 Slide 6 of

27 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review TENSORS Kronecker s Delt ij = ji [I] = [I] T ij V j = V i [I] {V}= {V} ij V jk = V ik ij V ij = V ii ij ij = ii = Revised /0 Slide 7 of

28 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review TENSORS (e T ) (e ) (e T T xx xy xz xy yy yz ) xz yz zz Revised /0 Slide 8 of

29 Revised /0 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review Slide 9 of With the ngles etween nd the xes, F k j i F k j i F k F j F i F F F F F F F F z y x z y x z y x z z y y x x cos cos cos cos cos cos cos cos cos RECTNGULR COMPONENTS

30 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review RECTNGULR COMPONENTS - 0 Revised /0 is unit vector long the line of ction of F nd cos x, cos re the direction cosines for F y, nd cos z Slide 0 of

31 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review MTRIX NOTTION OF VECTORS Row Vector u u,,,, u u u u u Column Vector v v v v Revised /0 Slide of

32 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review MTRIX NOTTION OF VECTORS u u, u, u T u u u u Revised /0 Slide of

33 Revised /0 4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review Slide of ) ( ) ( ) ( det v u u v e v u u v e v u v u e v v v u u u e e e v u v u u v v u v u v u VECTOR DOT & CROSS PRODUCTS