ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR AGEBRA Mtrices nd Vectors Prof. Dr. Bülent E. Pltin Spring Sections & /
ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR AGEBRA Mtrices nd Vectors Mtri: A rectngulr rry of sclrs (numbers, vribles, or functions rel or comple). M m M m O n n M mn [ ] [ A] ij Rows Elements ; i,,...,m, j,,...,n Columns Totl no of rows Element i,j i th row j th column Size or dimension of mtri: m n Totl no of columns Prof. Dr. Bülent E. Pltin Spring Sections & /
ME Applied Mthemtics for Mechnicl Engineers Vector: A mtri with only one row (size n, row vector) or only one column (size m, column vector). Equlity of mtrices: Two mtrices nd [B] re sid to be equl to ech other if nd only if i) they hve the sme dimension m n, nd ii) their corresponding elements re equl; i.e., ij b ij for ll i,,, m nd j,,, n Addition/Subtrction of mtrices: Addition/subtrction is defined only for mtrices of the sme size nd result in nother mtri of the sme size. For two mtrices nd [B] of the sme size [C] ± [B] implies tht c ij ij ± b ij for ll i,,, m nd j,,, n Prof. Dr. Bülent E. Pltin Spring Sections & /
ME Applied Mthemtics for Mechnicl Engineers Emple: Given 6 nd [B] 6 6 [ A] [ B] 6 6 9 7 [ A] - [ B] Multipliction/Division by sclr: Multipliction/division of mtri by sclr implies tht ll its elements re to be multiplied/divided by the sme sclr ; i.e., [ ij ] or / [ ij /] Prof. Dr. Bülent E. Pltin Spring Sections & /
ME Applied Mthemtics for Mechnicl Engineers Emple: Given.7 -.8.9.6 Then..8.6 7. nd 9 Some importnt properties of mtrices: Given mtrices, [B], nd [C] of the sme size nd set of sclr constnts,, nd, the following properties hold: [B] [B] commuttive ([B] [C]) ( [B]) [C]) ssocitive ( [B]) [B] distributive ( ) ( ) ( ) ( ) Prof. Dr. Bülent E. Pltin Spring Sections & /
ME Applied Mthemtics for Mechnicl Engineers Mtri multipliction: et be m n mtri nd [B] be p q mtri. The mtri product [B] is defined only if n p nd it gives mtri [C] of size m q nd shown s [C] [B] The elements of [C] re given s c ij i b j i b j.. in b nj Note tht the element c ij cn be interpreted s the dot product (inner product) of the i th row vector of nd the j th column vector of [B]. [B] n i b j Prof. Dr. Bülent E. Pltin Spring Sections & 6/
ME Applied Mthemtics for Mechnicl Engineers Emple: Given 6 nd [B] 6 Since number of columns of nd number of rows of [B] mtch, nd [B] mtrices re clled comptible s fr s [B] opertion is concerned. Hence, their product becomes 7 6 [ C] [ A][ B] 6 6 6 6 6 Prof. Dr. Bülent E. Pltin Spring Sections & 7/
ME Applied Mthemtics for Mechnicl Engineers Note tht, in this emple, even though [B] multipliction is defined, [B] multipliction is not defined since the number of columns of [B] (which is ) nd number of rows of (which is ) do not mtch. Mtri multipliction is not commuttive. Even in cses where both [B] nd [B] multiplictions re defined, these multiplictions re usully not equl to ech other; i.e., [B] [B], in generl Note tht only if the mtri is of size m n nd [B] is n m, then both [C] [B] nd [D] [B] re defined where [C] nd [D] re squre mtrices of different sizes m nd n, respectively; therefore, not equl to ech other. If mtrices nd [B] re both squre nd of the sme size n, then both [C] [B] nd [D] [B] re defined where [C] nd [D] re squre mtrices of the sme size n; but not necessrily equl to ech other. Prof. Dr. Bülent E. Pltin Spring Sections & 8/
ME Applied Mthemtics for Mechnicl Engineers Prof. Dr. Bülent E. Pltin Spring Sections & 9/ Emple: Trnspose of mtri: The trnspose of n m n mtri is n n m mtri [B] whose rows re the columns nd columns re the rows of. It is denoted by T. Hence, [ ij ] T [ ji ] Given squre mtrices [B] nd 6 [B] 7 [B] Emple: Given squre mtrices [B] nd [B] [B]
ME Applied Mthemtics for Mechnicl Engineers Emple: 8 8 7 T [b] [b] T [ 7 ] Note tht [b] T is convenient wy to describe row vector. The following re two importnt properties of trnspose opertion. ( [B]) T T [B] T ( [B]) T [B] T T Note the chnge of the order Prof. Dr. Bülent E. Pltin Spring Sections & /
ME Applied Mthemtics for Mechnicl Engineers Prof. Dr. Bülent E. Pltin Spring Sections & / Emples of sprse mtrices: Squre mtri: A mtri with equl number of rows nd columns. Principl (min) digonl Off-digonl elements [C] [B] Sprse mtri: A mtri with most of its elements zero [ ] nn n n n n A M O M M
ME Applied Mthemtics for Mechnicl Engineers Symmetric mtri: A squre mtri whose off digonl elements t symmetric loctions re equl; Emple: tht is ij ji for ll i, j (,,, n) Note tht the trnspose of symmetric mtri is equl to itself; i.e., T Sew-symmetric mtri: A squre mtri whose off digonl elements t symmetric loctions re Emple: equl in size but of opposite sign; tht is ij ji for ll i, j (,,, n) Note tht the elements of min digonl of [B] sew-symmetric re ll zero; i.e., ii for ll i,,, n Prof. Dr. Bülent E. Pltin Spring Sections & /
ME Applied Mthemtics for Mechnicl Engineers Prof. Dr. Bülent E. Pltin Spring Sections & / Note tht ny squre mtri,, cn be written s s s where is clled the symmetric prt of, nd is clled the sew-symmetric prt of. T s T s Emple: 7 7 7 s 7 7 s
ME Applied Mthemtics for Mechnicl Engineers Prof. Dr. Bülent E. Pltin Spring Sections & / Tringulr mtri: A squre mtri whose off digonl elements bove or below the min digonl re ll zero. Emple: Upper tringulr mtri ower tringulr mtri Digonl mtri: A squre mtri whose off digonl elements re ll zero. Tht is ij for ll i, j,,, n, i j [B]
ME Applied Mthemtics for Mechnicl Engineers Prof. Dr. Bülent E. Pltin Spring Sections & / Identity (unity) mtri: A digonl mtri whose elements in the principl digonl re ll. Tht is ii for ll i,,, n Null mtri: A m n mtri whose elements re ll zero. Tht is, ij for ll i,,, m nd j,,, n [B] [C] [B]
ME Applied Mthemtics for Mechnicl Engineers Prof. Dr. Bülent E. Pltin Spring Sections & 6/ Note tht, contrry to the cse in multiplictions of sclrs, if the multipliction of two squre mtrices gives null mtri then this does not imply tht t lest one of the mtrices multiplied should be null mtri s well. Bnded mtri: A squre mtri; some of its digonls net to the min digonl re not zero, but the rest of the off-digonl elements re ll zero. [B] but [B] [B] &
ME Applied Mthemtics for Mechnicl Engineers Prof. Dr. Bülent E. Pltin Spring Sections & 7/ Emple: Consider the free (unforced) motion of the following model which my be representing the longitudinl vibrtions of long elstic body whose both ends re fied. Using the Newton s nd lw of motion for ech mss, the governing equtions of motion cn be written s: 6 m m m m m ) ( m ) ( ) ( m ) ( ) ( m ) ( ) ( m ) ( m 6 && && && && &&
ME Applied Mthemtics for Mechnicl Engineers Prof. Dr. Bülent E. Pltin Spring Sections & 8/ [ ] [ ] [ ] [ ] [] M K && Mss mtri digonl Position vector Stiffness mtri, symmetricl nd bnded Forcing vector null vector Governing eqution in mtri form: m m m m m [M] [] 6 [K] []
ME Applied Mthemtics for Mechnicl Engineers Determinnts, Minors, nd Cofctors Determinnt: It is sclr quntity nd defined only for squre rrys. Every squre rry A of size n hs unique determinnt vlue D nd it is shown s D M n M n O n n M nn A det Prof. Dr. Bülent E. Pltin Spring Sections & 9/
ME Applied Mthemtics for Mechnicl Engineers Minor: It is determinnt of n rry, which is one lower size thn the rry of n originl determinnt. A minor is lwys ssocited with only one of the elements of n originl rry. Its rry is obtined by deleting the row nd column of the originl rry contining tht prticulr element; tht is the minor M ij of the element ij of n rry of size n is the determinnt of the rry of size n, which is formed by deleting the i th row nd j th column of the originl rry. Emple: A M M Prof. Dr. Bülent E. Pltin Spring Sections & /
ME Applied Mthemtics for Mechnicl Engineers Cofctor: It is lso ssocited with only one of the elements of n originl rry nd defined s C ij ( ) ij M ij Emple: A C ( ) M C ( ) M Evlution of Determinnts: The vlue of the determinnt of size one is sme s the sclr involved; tht is The vlue of the determinnt of size two is obtined s Prof. Dr. Bülent E. Pltin Spring Sections & /
ME Applied Mthemtics for Mechnicl Engineers Determinnts of size three nd higher cn be evluted by epnding it with respect to one of its rows or columns s D M n M n O n n M nn n j ny i ij C ij n i ny j ij C ij Emple: 6 6 D 6 ( ) ( ) ep nsion w. r. t. st row Prof. Dr. Bülent E. Pltin Spring Sections & /
ME Applied Mthemtics for Mechnicl Engineers END OF WEEK Prof. Dr. Bülent E. Pltin Spring Sections & /