Nothing Special   »   [go: up one dir, main page]
Scribd Logo
Upload

Welcome to Scribd!

  • 147 views

    Engineering Mathematics PDF

    Uploaded by

    prajakta chaudhari
    The document defines and provides examples of 18 types of matrices: 1) Row matrix and column matrix 2) Null or zero matrix 3) Square matrix 4) Diagonal matrix 5) Scalar matrix 6) Unit or identity matrix 7) Upper triangular matrix and lower triangular matrix 8) Transpose of a matrix 9) Symmetric matrix and skew-symmetric matrix 10) Orthogonal matrix 11) Hermitian matrix and skew-hermitian matrix 12) Unitary matrix 13) Normal matrix 14) Singular matrix and non-singular matrix

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd

    Engineering Mathematics PDF

    Uploaded by

    prajakta chaudhari
    147 views4 pages
    The document defines and provides examples of 18 types of matrices: 1) Row matrix and column matrix 2) Null or zero matrix 3) Square matrix 4) Diagonal matrix 5) Scalar matrix 6) Unit or identity matrix 7) Upper triangular matrix and lower triangular matrix 8) Transpose of a matrix 9) Symmetric matrix and skew-symmetric matrix 10) Orthogonal matrix 11) Hermitian matrix and skew-hermitian matrix 12) Unitary matrix 13) Normal matrix 14) Singular matrix and non-singular matrix

    Original Description:

    Original Title

    ENGINEERING MATHEMATICS.pdf

    Copyright

    © © All Rights Reserved

    Available Formats

    PDF, TXT or read online from Scribd

    Did you find this document useful?

    Is this content inappropriate?

    The document defines and provides examples of 18 types of matrices: 1) Row matrix and column matrix 2) Null or zero matrix 3) Square matrix 4) Diagonal matrix 5) Scalar matrix 6) Unit or identity matrix 7) Upper triangular matrix and lower triangular matrix 8) Transpose of a matrix 9) Symmetric matrix and skew-symmetric matrix 10) Orthogonal matrix 11) Hermitian matrix and skew-hermitian matrix 12) Unitary matrix 13) Normal matrix 14) Singular matrix and non-singular matrix

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    147 views4 pages

    Engineering Mathematics PDF

    Uploaded by

    prajakta chaudhari
    The document defines and provides examples of 18 types of matrices: 1) Row matrix and column matrix 2) Null or zero matrix 3) Square matrix 4) Diagonal matrix 5) Scalar matrix 6) Unit or identity matrix 7) Upper triangular matrix and lower triangular matrix 8) Transpose of a matrix 9) Symmetric matrix and skew-symmetric matrix 10) Orthogonal matrix 11) Hermitian matrix and skew-hermitian matrix 12) Unitary matrix 13) Normal matrix 14) Singular matrix and non-singular matrix

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 4

    MPSC Civil Engg.

    Pre Exam – ENGINEERING MATHEMATICS Page | 1

    ENGINEERING MATHEMATI CS :

    Types of Matrices

    1. Row Matrix:
    A matrix having only one row is called as “Row Matrix or Row Vector”.
    e.g. [1 2 1],[1 0 2 3 −3],[𝑎] etc.
    2. Column Matrix :
    A matrix having only one column is called as “Column Matrix or Column vector”.
    1
    1
    e.g. :[2] ,[ ] ,[𝑏] etc.
    −1
    3
    Note that : A row or column matrix is also called as vector.
    3. Null or Zero Matrix:
    A matrix containing only zeroes is called as “Null or Zero Matrix”.
    0 0 0 [0]
    e.g. [ ] ,[ ] , etc.
    0 0 0
    4. Square Matrix:
    A matrix is said to be “Square Matrix”, if number of rows is equal to number of
    column in it.
    3 0 2
    1 0
    [ ] , [0 1 0] , [4]1×1 etc.
    0 1 2×2
    2 0 3 3×3
    5. Diagonal Matrix:
    A square matrix is said to be “Diagonal Matrix”, if all its non-diagonal entries are zero
    and atleast one diagonal element are non zero.
    1 0 0
    1 0
    e.g. [ ] , [0 0 0] , 𝑒𝑡𝑐
    0 5
    0 0 2
    6. Scalar Matrix :
    A diagonal matrix in which all entries on principal diagonal are equal is called as
    “Scalar Matrix”.
    2 0 0
    4 0
    e.g. : e.g. [ ] , [0 2 0] , 𝑒𝑡𝑐
    0 4
    0 0 2
    7. Unit Matrix or Identity Matrix:
    A scalar matrix in which all entries on principal diagonal are unit (one) is called as
    “Unit or Identity Matrix”.
    1 𝑖𝑓 𝑖 = 𝑗
    i.e., [𝑎𝑖𝑗 ] is a Identity, if 𝑎𝑖𝑗 ={
    0 𝑖𝑓 𝑖 ≠ 𝑗

    Youtube: Infinity Academy t.me/infinitympsccivil 7887811411


    MPSC Civil Engg. Pre Exam – ENGINEERING MATHEMATICS Page | 2

    1 0 0
    1 0
    e.g. [ ] , [0 1 0] etc
    0 1
    0 0 1
    8. Upper Triangular Matrix:
    A matrix in which all entries below principal diagonal are zero is called as “Upper
    triangular matrix”.
    i.e., [𝑎𝑖𝑗 ] is a upper triangular, iff 𝑎𝑖𝑗 = 0 𝑖𝑓 𝑖 > 𝑗
    1 2 3
    1 2
    e.g. :[ ] , [0 4 5] etc.
    0 3
    0 0 6

    9. Lower Triangular Matrix:


    A matrix in which all entries above principal diagonal are zero is called as “Lower
    triangular matrix”.
    i.e., [𝑎𝑖𝑗 ] is a lower triangular, iff 𝑎𝑖𝑗 = 0 𝑖𝑓 𝑖 < 𝑗
    • Transpose of Matrix: The matrix obtained by inter changing rows in to column (or
    column in to rows), is called as transpose of given matrix. If A is given matrix then
    it’s transpose is denoted by 𝐴𝑇 (𝑜𝑟 𝐴𝑡 𝑜𝑟 𝐴′).
    i.e., If A = [𝑎𝑖𝑗 ] then 𝐴𝑇 = [𝑎𝑖𝑗 ]
    𝑎 𝑏 𝑎 𝑐
    e.g. : If A =[ ], then 𝐴𝑇 = [ ]
    𝑐 𝑑 𝑏 𝑑
    • Real Matrix : Any matrix with all its entries as a real numbers is called as “Real
    matrix”
    • Complex Matrix : Any matrix with its entries as a complex numbers is called as
    “Complex matrix”
    10. Symmetric Matrix :
    A square matrix A is said to be “Symmetric Matrix”
    if = 𝐴𝑇 . i.e., If A is symmetric then A = [𝑎𝑖𝑗 ] = [𝑎𝑗𝑖 ] = 𝐴𝑇
    𝑎 ℎ 𝑔
    e. g. 𝐴 = [ℎ 𝑏 𝑓 ] , 𝐵 = [2 3
    ] = 𝐵𝑇 , etc.
    3 4
    𝑔 𝑓 𝑐

    11. Skew-symmetric Matrix:


    A square matrix A is said to be “Skew-symmetric Matrix”,
    if = − 𝐴𝑇 . i.e., If A is skew-symmetric then A = [𝑎𝑖𝑗 ] = [𝑎𝑗𝑖 ] = −𝐴𝑇
    0 1 2
    0 −𝑎
    e. g. 𝐴 = [−1 0 3] , 𝐵 = [ ], etc.
    𝑎 0
    −2 −3 0
    Note that :
    • In a skew-symmetric matrices all diagonal entries are always zero.
    Since, for diagonal entries 𝑎𝑖𝑖 = −𝑎𝑖𝑖 which is possible only if 𝑎𝑖𝑖 = 0

    Youtube: Infinity Academy t.me/infinitympsccivil 7887811411


    MPSC Civil Engg. Pre Exam – ENGINEERING MATHEMATICS Page | 3

    • If A is any square matrix, 𝐴 + 𝐴𝑇 is always symmetric while 𝐴 − 𝐴𝑇 is always skew-


    𝐴+𝐴𝑇 𝐴−𝐴𝑇
    symmetric, moreover we can write 𝐴 = 2
    + 2
    . i.e., any square matrix can
    be represented into the sum of symmetric and skew-symmetric matrices.
    12. Orthogonal Matrix
    A square matrix A is said to be “Orthogonal Matrix”,
    if 𝐴 𝐴𝑇 = I = 𝐴𝑇 A or 𝐴𝑇 = 𝐴−1
    Note That:
    (a) If A is an Orthogonal Matrix then, 𝐴−1 =𝐴𝑇 .
    (b) If A is an Orthogonal Matrix then, = |𝐴| = ± 1 .
    (c) If A and B are an Orthogonal Matrices then, A, 𝐴−1 , 𝐴𝑇 and AB are also
    Orthogonal.
    (d)If A is real matrix, then (𝑖) A is orthogonal, (𝑖𝑖) the rows of A forms an orthonormal
    set and (𝑖𝑖𝑖) the columns of A forms an orthonormal set, are equivalent.
    (e) If A and B are symmetric matrices, then 𝐴 + 𝐵, 𝑘𝐴 (𝑓𝑜𝑟 𝑎𝑛𝑦 𝑠𝑐𝑎𝑙𝑎𝑟 𝑘), 𝐴2 , 𝐴𝑛 (𝑓𝑜𝑟 𝑛 >
    0) are also symmetric
    13. Hermitian matrix
    A square complex matrix A is said to be “Hermitian Matrix”
    ̅̅̅̅
    If 𝐴 = 𝐴𝐻 , where 𝐴𝐻 = (𝐴 𝑇 ) 𝑜𝑟 𝐴𝐻 = (𝐴̅)𝑇

    0 1 + 2𝑖 2 − 3𝑖
    𝑎 𝑐 + 𝑑𝑖
    e. g. 𝐴 = [−1 + 2𝑖 0 3 + 4𝑖 ] = −𝐴𝐻 , 𝐵 = [ ] = 𝐵𝐻 , 𝑒𝑡𝑐.
    𝑐 − 𝑑𝑖 𝑏
    2 + 3𝑖 −3 + 4𝑖 0
    14. Skew – herimition Matrix:
    A square complex matrix A is said to be “Skew – Herimitian matrix”
    If 𝐴 = −𝐴𝐻 , where 𝐴𝐻 = ̅̅̅̅
    𝐴𝑇
    0 1 + 2𝑖 2 − 3𝑖
    𝑎 𝑐 + 𝑑𝑖
    e. g. 𝐴 = [−1 + 2𝑖 0 3 + 4𝑖 ] = −𝐴𝐻 , 𝐵 = [ ( ] = −𝐵𝐻 , 𝑒𝑡𝑐.
    − 𝑐 − 𝑑𝑖 ) 𝑏
    −2 + 3𝑖 −3 + 4𝑖 0
    Note that:
    1. In a Hermitian matrices all diagonal entries are always Real.
    2. In a skew-hermitian matrices all diagonal entries are always zero.
    3. If matrix A is real matrix, then 𝐴𝑇 = 𝐴𝐻 .
    4. If A is any complex square matrix, then 𝐴 + 𝐴𝐻 is always Herimitian and skew –
    herimitian matrices.
    15. Unitary Matrix :
    A square complex matrix A is said to be “Unitary Matrix”,
    if 𝐴 𝐴𝐻 = 𝐼 = 𝐴𝐻 𝐴 𝑜𝑟 𝐴𝐻 = 𝐴−1 .
    Note that, (1) If A and B are unitary, then 𝐴𝐻 , 𝐴−1 . and AB are also unitary.
    (2) If A is complex matrix, then (𝑖) A is unitary, (ii) the rows of A forms an orthonormal
    set and

    Youtube: Infinity Academy t.me/infinitympsccivil 7887811411


    MPSC Civil Engg. Pre Exam – ENGINEERING MATHEMATICS Page | 4

    (iii) the columns of A forms an orthonormal set, are equivalent.


    16. Normal Matrix:
    A square real (or Complex) matrix A is said to be “ Normal Matrix”, if its commutes
    with 𝐴𝑇 (or 𝐴𝐻 ). i. e. 𝐴𝐴𝑇 = 𝐴𝑇 𝐴 ( 𝑜𝑟 𝐴𝐴𝐻 = 𝐴𝐻 𝐴)
    2 1 2 1 2 −1 5 0
    Ex: Let A = [ ] , then 𝐴 𝐴𝑇 = [ ][ ]=[ ]
    −1 2 −1 2 −1 2 0 5
    2 1 2 −1 5 0
    and 𝐴𝑇 𝐴 = [ ][ ]=[ ]
    −1 2 −1 2 0 5
    Hence, A is Normal.
    17. Singular matrix:
    A square matrix A is said to be “ Singular matrix” , |𝐴| = 0.
    2 1
    e. g. The matrix A = [ ] is singular, since |𝐴| = 0
    4 2
    1 3 1
    The matrix B = 0 1 0] is singular Since |𝐵| = 0
    [
    2 0 2
    18. Non-singular Matrix :
    A square matrix A is said to be “Non-singular Matrix”, if |𝐴| ≠ 0
    2 1
    e.g. : The matrix A = [ ] is non-singular, since |𝐴| ≠ 0
    5 3
    1 3 1
    Similarly the matrix 𝐵 = [0 1 0]is non-singular.
    2 0 4

    Youtube: Infinity Academy t.me/infinitympsccivil 7887811411

    You might also like