Matrix Theory PDF

Theorem 2 expressed as asumnofa Every square matrix can matrix. be uniquely symmetric: matrix andaskew-symmetric Proof: matrix. Then by the above theorem A+A' is...

Theorem 2 expressed as asumnofa Every square matrix can matrix. be uniquely symmetric: matrix andaskew-symmetric Proof: matrix. Then by the above theorem A+A' is Let Abe asquare and-isskew symmetric. symmetric Hence expressed Aas the sum ofasymmetric and a skew-Symmetric Thus we have matrix. Ifpossiblelet A= P+0 (1) where Pis symmetric and Qisskew-symmetric. Now =(P+0)' =pl+0'- P-9.. (2) Solving for Pand Qfrom (l) and (2) we get aQ-4-4) Hence the representation ofAas the sum ofa symmetric matrix and aSkew Symmetric matrix is unique. Theorem 3 A IfA is asquare matrix, then A+4' is Hermitian skew-Hermitian and -T Proof: Now -A+4 srices Hermitian,.4+4' isa 0,2,6270 80, 3hb5.7--7-A -A is skew- Hermitian Theorem 4 ysquare matrix is uniquely expressed as asum ofa Hermitian matrix and Iskew-Hermitian matrix. Proof: LetA be asquare matrix. W have seen that 4+4 is Hermitian and 4-A isskew-Hermitian. Hence Thus Acan be expressed as asum ofHermitian matrix and aSkew-Hermitian matrix. Ifpossible let A= R+.. (1) where Pis a Hermitianmatrix and Ois Skew Hermitian. A-(R+)' -R +0 - R-9 (2) From (1) and(2) H+)=R amd Hence the representation of Ais unique. Singular matrix : Asquare matrix Ais saidto be singular if A is zero. ie., determinant Ais Zero.  Non-singularmatrix : Asquare matrix Ais said to be non-singular if det A 0 i.e., lAlz0 Inverse ofa matrix : such that Iftwo square matrices Aand Bare of same order and AB = then A is called tl.e inverse of B and vice versa. The inverse of Ais BA=1, denoted by The following facts of invèrse caneasily be proved. 1. The inverse ofa matrix is unique. 2 The inverse of amatrix exists ifand only ifA is non-Singular. 3. If 4and B are non-singular square matrices of same order then (AB) =Ba 4. $6.5 Orthogonal matrix : Asquare matrix Ais orthogonal A4=A'A =I Now AA =I This gives 44-l1 ie, 1al4=1 14f=1 (=4) : |4|=+1.. A is anon-singular matrix and hence - exists and A'=4 $66Unitarymatrix: fis an Asquare matrix A unitary matrix if AA =I ie., AA =1 Example4 611 TheoremS matrices of the same order, then ABand BA are AIfA and Bare orthogonal als0 orthogonal. Proof: matrices Given Aand Bareorthogonal (1) (2) BB= BB=l Now (4B)(48)-AB(B4") since matrix multiplication is associative - AlA =A#using (1) and (2) Similarly we can prove that (AB)'(AB) =l :: AB is orthogonal. Alo (BA)(B4 -(24)(4 8') - BA4")B= BIB' =BB' =l :. BA is also orthogonal. Theorem 6 lfA is orthogonal matrix, showthat 4 and - are also orthogonal matrices. Proof: Since Ais orthogonal AA' =1 Taking transpose on both sides :. A' is orthogonal AgainA=I d $6.10 Eigen Values and Eigen Vectors: L A-(a,)be asquare matrix of ordér n. Characteristic value problem is in vhich we have to find the scalar A and non-zero vectors X=). satisfying theequation AX= X or (4-A) X=0, /being the unit matrix of order n. For the abovesystem of equations to have non-trivial solution the condition sJ4-=0, The equation 4-=0 is called the characteristic équation. The roots of the equation |4-|=0 are the eigen values or latent roots or characteristic values ofA. The corresponding non-zero vectors X toeach eigen valueAsatisfying the equation (A- ) X=0 are called the eigen vectors or charácteristic vectors of the matrix A. $6.11 Cayley - Hamilton Theorem: Every square matrix satisfies its own characteristic equation. SShort cutmethod to find the characteristic equation of agiven 3x3 matrix A: IfA is a 3x3 matrix, its characteristic equation is given by P-a, +a,l -a, =0 where a, =Sum of leading diagonal elements. elements. a,=Sum ofminors of leading diagonal a =Determinant of A. $Diagonalisation of Matrices means finding another matrix Diagonalisingthe g'AB matrix is a A' diagonal matrix. B(calledtheMODAL. matrix)suchthat matrixA: diagonalise a Workingruleto ofA and obtain the Eigen characteristic equation 1 Formthe corresponding to the Eigen values values / sof A. 2 Find the Eigen vectors columns are the Eigen vectors of 3 Form the matrix Bwhose -!. 4. Findthe inverse ofB. i.e., Find This will be a diagonal matrix whose whose diagonal 5. Find R-1 AB. Eigen values of 4. elements are Note 1. We can diagonalise a matrix A if all the Eigen values are distinet 2 Ifthe Eigen values are not distinct, it may not be possible to diagonalise A.