`text(Matrix Polynomial :)` `f(x)=a_0x^n+a_1x^(n-1)+a_2x^(n-2)+............+a_nx^0` then we define a matrix polynomial
`f(A)=a_0A^n+a_1A^(n-1)+a_2A^(n-2)+.............+a_nI_n`
where `A` is the given square manix. lf `f(A)` is the null manix then `A` is called the zero or root of the manix polynomial `f(x)`.
Note that `(A)^0` is not defined if `A` is a null matrix.
Definitions :
`text(Unit or Identity Matrix :)`
A diagonal matrix is said to be an identity matrix, if its diagonal elements are equal to 1.
`a_(ij) = { tt ((0, if i!=j),(1, if i=j))`
`text(Triangular Matrix :)`
A square matrix is called a triangular matrix, if its each element above or
below the principal diagonal is zero. It is of two types:
`(i)` `text(Upper Triangular Matrix)` A square matrix in which all elements below
the principal diagonal are zero is called an upper triangular matrix i.e., a
matrix `A= [a_(ij) ]_( m xx n)` is said to be an upper triangular matrix, if `a_(ij) = 0,`
when `i > j.`
For example,
`(i) [(3,-2,4,1),(0,2,-3,2),(0,0,0,8)]_(4 xx 4)` `(ii) [(a_11 , a_12 , a_13, a_14 , a_15 ),(0, a_22 , a_23 , a_24 , a_25),(0 , 0, a_33 , a_34 , a_35),(0 ,0 ,0,a_44 , a_45),(0,0,0,0,a_55)]_(5xx5)`
are upper triangular matrices.
(ii) Lower Triangular Matrix A square matrix in which all elements above
the principal diagonal are zero is called a lower triangular matrix i.e., a
matrix `A= [a_(ij)(n xx n)` is said to be a lower triangular matrix, if aij = 0,
when `i < j.`
For example , `(i) [(7,0,0),(5,4,0),(2,3,4)_(3xx 3)]` `(ii)[(10 , 0 ,0 , 0),(8,9,0,0),(5,6,7,0),(1,2,3,4)]_(4 xx 4)`
are lower triangular matrices.
`text(Note)` Minimum number of zeroes in a triangular matrix is given by `(n(n- 1))/2`, where n
is order of matrix.
` text( Diagonal Matrix :)`
A square matrix is said to be a diagonal matrix, if all its non-diagonal elements are zero. Thus, `A= [a_(ij)]_(nxxn),` is called a diagonal matrix, if `a_(ij) = 0,` when `i !=j`
e.g. `A = [2]`
`A = [(-1,0),(0,2)]`
`(I) text(Scalar Matrix :)`
A diagonal matrix is said to be a scalar matrix, if its diagonal elements are equal. Thus, `A= [a_(ij)]_(nxxn)`, is called scalar matrix, if
`a_(ij) = { tt((0, if i!=j),(k, if i=j))`
e.g. `A = [(2,0),(0,2)]`
`text( Idempotent Matrix :)`
A square matrix is idempotent provide `A^2 = A` . For an idempotent matrix
`A, A^n=A forall n ge 2,n in N => A^n=A,n ge 2`.
For example if `A=[(2,-2,-4),(-1,3,4),(1,-2,-3)]` then `A^2=A` i.e. `A` is idempotent.
`text( Nilpotent Matrix :)`
A square matrix is said to be nilpotent matrix of index `p, (p in N)`, if `A^P = O , A^(p-1) ne O` i.e. if `p` is the least positive integer for which `A^P = O` , then `A` is said to be nilpotent of index `p`.
e.g. (i) `A=[(1,1,3),(5,2,6),(-2,-1,3)]` Note that `A^3 = 0` but `A^2 ne 0 =>` index of nilpotency `= 3`
(ii) `A=[(ab,b^2),(-a^2,-ab)]` is a nilpotent matrix of index `2` .
(iii) `A=[(a,-a^2),(1,-a)][(a,-a^2),(1,-a)]` nil potent.
`text(Periodic Matrix :)`
A square matrix which satisfies the relation `A^(K+1) =A`, for some positive integer `K` then `A` is periodic with period K i.e. if `K` is the least positive integer for which `A^( K+1) = A` then `A` is said to be periodic with period `K`. If `K = 1` then `A` is called idempotent.
e.g. the matrix `[(2,-3,-5),(-1,4,5),(1,-3,-4)]` has the period `1`.
Note : (1) Period of a square null matrix is not defined.
(2) Period of an idempotent matrix is `1`.
`text( Involutory Matrix :)`
If `A^2 = 1` , the matrix is said to be an involutary matrix. An involutary matrix is its own inverse.
e.g. (i) `A=[(0,1),(1,0)][(0,1),(1,0)]=[(1,0),(0,1)]`;
`text(Orthogonal Matrix)`
A square matrix A is said to be orthogonal matrix, iff AA' = I, where 1 is an
identity matrix.
Note 1 If `AA' =I` , then `A^(-1) =A.`
Note 2 If A and Bare orthogonal, then AB is also orthogonal.
Note 3 If A is orthogonal, then `A^(-1)` and A' are also orthogonal.
`text(Illustration)` If `A = [(1,2,2),(2,1,-2),(a,2,b)]` is a matrix satsfying `AA' = 9I_3,` then
find the value of `| a | + | b |.`
`text(Solution)` Since, `AA' = 9 I_3`
`=> [(1,2,2),(2,1,-2),(a,2,b)] [(1,2,a),(2,1,2),(2,-2,b)] = 9 [(1,0,0),(0,1,0),(0,0,1)]`
`=> [(9,0,a+2b+4),(0,9,2a-2b+2),(a+2b+4 , 2a -2b +2 , a^2+b^2 + 4)] = [(9,0,0),(0,9,0),(0,0,9)]`
Equating the corresponding elements, we get
`a+ 2b + 4 =0...............(i)`
`2a-2b+2=0.....................(ii)`
and `a^2 + b^2 + 4 = 9.................(iii)`
From Eqs. (i) and (ii), we get
`a = - 2` and `b = - 1`
Hence, `| a| +| b| = |-·2| +|-1|= 2 + 1 = 3`
`text(Matrix Polynomial :)` `f(x)=a_0x^n+a_1x^(n-1)+a_2x^(n-2)+............+a_nx^0` then we define a matrix polynomial
`f(A)=a_0A^n+a_1A^(n-1)+a_2A^(n-2)+.............+a_nI_n`
where `A` is the given square manix. lf `f(A)` is the null manix then `A` is called the zero or root of the manix polynomial `f(x)`.
Note that `(A)^0` is not defined if `A` is a null matrix.
Definitions :
`text(Unit or Identity Matrix :)`
A diagonal matrix is said to be an identity matrix, if its diagonal elements are equal to 1.
`a_(ij) = { tt ((0, if i!=j),(1, if i=j))`
`text(Triangular Matrix :)`
A square matrix is called a triangular matrix, if its each element above or
below the principal diagonal is zero. It is of two types:
`(i)` `text(Upper Triangular Matrix)` A square matrix in which all elements below
the principal diagonal are zero is called an upper triangular matrix i.e., a
matrix `A= [a_(ij) ]_( m xx n)` is said to be an upper triangular matrix, if `a_(ij) = 0,`
when `i > j.`
For example,
`(i) [(3,-2,4,1),(0,2,-3,2),(0,0,0,8)]_(4 xx 4)` `(ii) [(a_11 , a_12 , a_13, a_14 , a_15 ),(0, a_22 , a_23 , a_24 , a_25),(0 , 0, a_33 , a_34 , a_35),(0 ,0 ,0,a_44 , a_45),(0,0,0,0,a_55)]_(5xx5)`
are upper triangular matrices.
(ii) Lower Triangular Matrix A square matrix in which all elements above
the principal diagonal are zero is called a lower triangular matrix i.e., a
matrix `A= [a_(ij)(n xx n)` is said to be a lower triangular matrix, if aij = 0,
when `i < j.`
For example , `(i) [(7,0,0),(5,4,0),(2,3,4)_(3xx 3)]` `(ii)[(10 , 0 ,0 , 0),(8,9,0,0),(5,6,7,0),(1,2,3,4)]_(4 xx 4)`
are lower triangular matrices.
`text(Note)` Minimum number of zeroes in a triangular matrix is given by `(n(n- 1))/2`, where n
is order of matrix.
` text( Diagonal Matrix :)`
A square matrix is said to be a diagonal matrix, if all its non-diagonal elements are zero. Thus, `A= [a_(ij)]_(nxxn),` is called a diagonal matrix, if `a_(ij) = 0,` when `i !=j`
e.g. `A = [2]`
`A = [(-1,0),(0,2)]`
`(I) text(Scalar Matrix :)`
A diagonal matrix is said to be a scalar matrix, if its diagonal elements are equal. Thus, `A= [a_(ij)]_(nxxn)`, is called scalar matrix, if
`a_(ij) = { tt((0, if i!=j),(k, if i=j))`
e.g. `A = [(2,0),(0,2)]`
`text( Idempotent Matrix :)`
A square matrix is idempotent provide `A^2 = A` . For an idempotent matrix
`A, A^n=A forall n ge 2,n in N => A^n=A,n ge 2`.
For example if `A=[(2,-2,-4),(-1,3,4),(1,-2,-3)]` then `A^2=A` i.e. `A` is idempotent.
`text( Nilpotent Matrix :)`
A square matrix is said to be nilpotent matrix of index `p, (p in N)`, if `A^P = O , A^(p-1) ne O` i.e. if `p` is the least positive integer for which `A^P = O` , then `A` is said to be nilpotent of index `p`.
e.g. (i) `A=[(1,1,3),(5,2,6),(-2,-1,3)]` Note that `A^3 = 0` but `A^2 ne 0 =>` index of nilpotency `= 3`
(ii) `A=[(ab,b^2),(-a^2,-ab)]` is a nilpotent matrix of index `2` .
(iii) `A=[(a,-a^2),(1,-a)][(a,-a^2),(1,-a)]` nil potent.
`text(Periodic Matrix :)`
A square matrix which satisfies the relation `A^(K+1) =A`, for some positive integer `K` then `A` is periodic with period K i.e. if `K` is the least positive integer for which `A^( K+1) = A` then `A` is said to be periodic with period `K`. If `K = 1` then `A` is called idempotent.
e.g. the matrix `[(2,-3,-5),(-1,4,5),(1,-3,-4)]` has the period `1`.
Note : (1) Period of a square null matrix is not defined.
(2) Period of an idempotent matrix is `1`.
`text( Involutory Matrix :)`
If `A^2 = 1` , the matrix is said to be an involutary matrix. An involutary matrix is its own inverse.
e.g. (i) `A=[(0,1),(1,0)][(0,1),(1,0)]=[(1,0),(0,1)]`;
`text(Orthogonal Matrix)`
A square matrix A is said to be orthogonal matrix, iff AA' = I, where 1 is an
identity matrix.
Note 1 If `AA' =I` , then `A^(-1) =A.`
Note 2 If A and Bare orthogonal, then AB is also orthogonal.
Note 3 If A is orthogonal, then `A^(-1)` and A' are also orthogonal.
`text(Illustration)` If `A = [(1,2,2),(2,1,-2),(a,2,b)]` is a matrix satsfying `AA' = 9I_3,` then
find the value of `| a | + | b |.`
`text(Solution)` Since, `AA' = 9 I_3`
`=> [(1,2,2),(2,1,-2),(a,2,b)] [(1,2,a),(2,1,2),(2,-2,b)] = 9 [(1,0,0),(0,1,0),(0,0,1)]`
`=> [(9,0,a+2b+4),(0,9,2a-2b+2),(a+2b+4 , 2a -2b +2 , a^2+b^2 + 4)] = [(9,0,0),(0,9,0),(0,0,9)]`
Equating the corresponding elements, we get
`a+ 2b + 4 =0...............(i)`
`2a-2b+2=0.....................(ii)`
and `a^2 + b^2 + 4 = 9.................(iii)`
From Eqs. (i) and (ii), we get
`a = - 2` and `b = - 1`
Hence, `| a| +| b| = |-·2| +|-1|= 2 + 1 = 3`