The Algebra (aljabr) of Matrices


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1 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering he lgebr (ljbr) of Mtrices lgebr s brnch of mthemtics is much broder thn elementry lgebr ll of us studied in our high school dys. In sense n lgebr is set of rules. s we lerned from our secondry eduction the lgebr of sclrs studies the opertions nd reltionships of sclr numbers. When the rules of ddition nd multipliction re generlized, their precise definitions led to the notions of lgebric structures tht cn led to the more esoteric concepts in sclr lgebr such s groups, rings nd fields fields of study in the relm of mthemtics known s bstrct lgebr, In tht sme sense we cn lso tlk in terms of the lgebr of comple numbers, lgebr of vectors, nd tensor lgebr. lgebr is set of definitions, rules, nd opertions tht govern mthemticl quntities.
2 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering Mtri Nottion Given the following system of equtions in short hnd the bove system cn be epressed s where n mn m m m n n n n b b b b m b b B mn n n n n n X X B
3 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering hus mtri is n ordered rrngement of vlues (sclr, vector, higher order tensor) in rowcolumn formt n n n n mn he mtri bove consists of m rows nd n columns. We cn identify the elements in mtri using the nottion ij. he first subscript designtes row in which the element is in nd the second subscript identifies the column. Repeted subscripts (indices) indictes the element is on the digonl. In lter courses the subscripted nottion will be used to represent the mtri itself when some rules re incorported on how to employ indicil nottion (see Elsticity or Continuum Mechnics).
4 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering lthough the number of rows nd columns of mtri my vry from problem to problem, two cses deserve ttention. When m = the mtri consists of one row of elements. his is clled row mtri nd is denoted B b b b m When n = the mtri consists of one column of elements nd is referred to s column mtri. It is denoted X n
5 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering Fundmentl ypes of Mtrices squre mtri hs the sme number of rows s columns. symmetric mtri is one in which the off digonl elements re reflected bout the digonl. Using subscript nottion ij ji i j or in full mtri formt 6 6 Symmetric squre mtrices ply specil role in engineering mthemtics. Cn mtri be symmetric if m does not equl n?
6 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering When ll elements of the min digonl re equl to one, nd ll the off digonl entries re equl to zero, i.e., I the squre mtri is referred to s the identity mtri. he trnspose of mtri is defined s the reordering of the elements of the mtri such tht the columns of the originl mtri become the rows of the new mtri. he following nottion is utilized n n n n mn n n n n mn 6
7 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering n emple of specific mtri nd its trnspose is 6 6 he product of trnspose is defined s For squre symmetric mtri B B ji ij i j 7
8 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering Show tht Proof by emple: If So tht B B B B Emple. 8
9 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering hen 8 8 B 8 B 9
10 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering Similrly hence B B B B
11 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering It is esy to conceptulize the trnspose of the trnspose is the mtri itself With rules of ddition nd multipliction B B B C B C nd n n where n
12 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering squre mtri is digonl mtri when the entries long the digonl re nonzero nd ll off digonl re zero, i.e., D d d dmm sclr mtri is digonl mtri whose digonl elements ll contin the sme sclr S I
13 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering he null mtri is mtri whose entries re ll zero, i.e., he null mtri does not hve to be squre mtri. he number of rows nd columns tht mtri hs is clled its order or its dimension. By convention, rows re listed first nd then columns. hus, we would sy tht the order (or dimension) of the mtri below is, mening tht the mtri hs rows nd columns
14 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering lower tringulr mtri is squre mtri with entries equl to zero bove the digonl, i.e., l l l L lm lm l mm n upper tringulr mtri is squre mtri with entries equl to zero below the digonl, i.e., u u u un u u u n u u unn
15 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering he mtri cn be dded to the mtri B to produce the mtri C Mtri ddition nd Subtrction ij m n B b ij m n C cijm n ij bij m n his points out tht to form sum of two mtrices the mtrices must be of the sme order (the mtrices re sid to be conformble for ddition) nd tht the elements of the sum re determined by dding the corresponding elements of the mtrices forming the sum.
16 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering Mtri ddition is both commuttive nd ssocitive he mtri {B} B B B C B C cn be subtrcted from the mtri {} B b ij m n ij m n to produce the mtri {C} o crry this one step further C cijm n ij bij m n B C C B 6
17 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering Mtri Multipliction We will see tht the mtri methods in structurl nlysis requires solving lrge systems of liner equtions using mtri lgebr tools. In n erlier section, the lrge systems of liner equtions ws represented simply s B X where ws n m n coefficient mtri, B ws n m vector, nd X ws n n vector. Now let m = n =. he mtrices tke on the following forms: b b b row by column element product nd summtion is clerly evident: 7
18 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering row by column element product nd summtion is clerly evident: b b b tht is, ech element of B is obtined by multiplying the corresponding element of by the pproprite element in X nd dding the result. Notice tht the forgoing procedure does not work if the number of columns in does not equl the number of rows of B. his suggests generl definition for the multipliction of two mtrices. If is n m n mtri, nd B is p q mtri, then eists if n is equl to p. B C 8
19 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering If this is the cse the elements of C re given by i,,,, m c ij m q n p k ik b kj j,,,, q Under these conditions the mtrices nd B re sid to be conformble for multipliction. In generl B B However, it cn be proven with some effort tht BC BC B C B BC BC C BC 9
20 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering Emple.
21 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering If the product of two mtrices nd B yields the null mtri, tht is, B it cnnot be ssumed tht or B is the null mtri. Furthermore, if or it cnnot be ssumed tht B C C B B his infers tht in generl cncelltion of mtrices in mnner similr to multipliction of sclr lgebr is not permissible. C
22 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering Emple.
23 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering Mtri Multipliction pplied to Structurl nlysis For displcement bsed structurl nlysis We will need mtri multipliction concepts etensively in order to formulte the epression bove. Lter we will solve this epression for {d}. n nn n n n d d d k k k k k k k f f f Stiffness mtri  Symmetric since k ij = k ji Nodl Displcements Nodl Forces
24 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering Powers nd Roots of Squre Mtrices Becuse the mtri is conformble with itself for mtri multipliction we cn form powers of the mtri s follows: n In ddition it is esy to see tht the lw of eponents holds m n ( n) m nd the zero power of mtri is the identity mtri. Negtive powers of mtri cn be defined s: m m nd re defined s: m m
25 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering Mtri Differentition One cn differentite mtri by differentiting every element of the mtri in the conventionl mnner. Consider he derivtive d[]/d of this mtri is d d
26 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering Similrly, one cn tke the prtil derivtive of mtri s follows In structurl nlysis we differentite strin energy potentil functions tht hve the form vi mtri multipliction y z z y z y z y y y z z y z y y y U y y U 6
27 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering Prtil differentition leds to Or in mtri formt If y y U U y U y y U X X y y U 7
28 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering hen U i X Here i represents nd y using inde nottion. he bove holds only if [] is symmetric mtri. 8
29 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering Mtri Integrtion One cn differentite mtri by differentiting every element of the mtri in the conventionl mnner. Consider he integrtion of this mtri is d We often integrte the epression X X d dy dz his triple product will be symmetric if [] is symmetric 9
30 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering Specil Ctegories of Mtrices squre mtri is sid to be skew mtri if ll digonl elements re not zero nd ij ji i j his mtri becomes skewsymmetric if ll digonl elements re zero. Here ny mtri cn be composed of comple elements C c ij j b j ij ij
31 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering comple mtri hs conjugte C c ij ij j b ij
32 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering Other Mtri erminology Bnded mtri If ll nonzero terms re contined within bnd long the digonl, the mtri is sid to be bnded 6 66 m, m mm, mm Lter you will find tht bnded mtrices re quite common in structurl nlysis. Specil bnd storge techniques re used to void finding spce for ll the zero entries.
33 Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering Sprse mtri If mtri hs reltively few nonzero terms (s is common in FE), the mtri is sid to be sprse Singulr mtri If the determinnt of the mtri equls zero, the mtri is sid to be singulr. s we sw erler, if [] is singulr, then the system of equtions []{}={b} hs no unique solution.
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