Matrices _ Class Notes
Page 3: Definition of Matrix
Matrix: A rectangular array of numbers arranged in rows and columns.
Order: If a matrix has order mxn, then it has m rows and n columns.
Total Elements: The number of elements in a matrix = mxn.
Square Matrix: A matrix where m=n (same number of rows and columns).
Page 4: Types of Matrices
Horizontal Matrix: Number of columns > number of rows.
Vertical Matrix: Number of rows > number of columns.
Examples:
Horizontal: [4 4 5 2 6 3 12]
Vertical: 2x3 matrix
E.g., 3 4 = 2x3
Page 5: Square Matrix Characteristics
Square Matrix: When m = n (e.g. 3x3).
Principal Diagonal: Elements where row index i equals column index j (elements a_ij where i=j).
Off-diagonal Elements: Elements where i ≠ j.
Count:
Elements on the principal diagonal = n.
Number of off-diagonal elements = n^2 - n.
Elements above and below the diagonal = n^2 - n/2.
Page 6: Types of Square Matrices
Diagonal Matrix: Matrix where all off-diagonal elements are zero; e.g., diag(1,0,2).
Triangular Matrix: Lower or upper matrices (either below or above the diagonal).
Product of all diagonal elements results in a determinant measure.
Page 7: Specific Matrix Types
Scalar Matrix: Diagonal matrix where all diagonal elements are equal.
Trace of a Square Matrix: Sum of all diagonal elements.
Unit Matrix/Identity Matrix: A square matrix with all diagonal entries as 1; denoted as I.
Trace Representation: Tr(A) = 1+2+1=4 for matrix A.
Page 8: Properties of Matrix Addition
Tr(A+B) = Tr(A) + Tr(B).
If O(A) = 1, O(B) = 1, define the addition A+B element-wise.
Scalar multiplication: KA = [kar_ij].
Example: If A is mxn, then det(KA) = k^n * det(A).
Page 9: Zeros in Matrices
For a triangular matrix (lower/upper), minimum zeros = n (Diagonal) - n + n = n(n-1)/2.
Addition of Matrices: A + B = B + A (commutative property).
Null Matrix: Denotes a zero matrix, [a_ij] where all elements = 0.
Page 10: Matrix Multiplication Basics
If A is mxn and B is nxp, then the product is defined as Crxp.
Matrix Multiplication: Conducted row by column.
Dimensions to note: mA > Columns, mB > Rows.
Page 11: Detailed Matrix Multiplication Process
R1C2, R1C3… = R2C1, …
Element (i,j) of AB = (i-th row of A) • (j-th column of B).
Example computations to illustrate matrix multipliers ABC...
Page 12: Determinants and Commutativity
Relation of determinants: det(AB) = det(A) * det(B).
For square matrices, det(A^n) = (det(A))^n.
Matrices A and B commute if AB = BA.
Anticommutative if AB = -BA.
Page 13: Nilpotent and Periodic Matrices
Nilpotent Matrix: If A^p = 0 for a smallest index p.
Periodic Matrix: A matrix is periodic if A^k = A for integer k.
Matrix classification: Singular if det(A)=0; Non-singular if det(A)≠0.
Page 14: Transpose Properties
Transpose of matrix A, denoted AT, obtained by interchanging rows and columns.
Dimension relation: O(A) = mxn, O(AT) = nxm.
Example of how elements shift when transposed.
Page 15: Symmetry in Matrices
Symmetric Matrix: A^T = A.
Skew-symmetric Matrix: A^T = -A (diagonal entries zero).
Importance of diagonal entry in determinants.
Page 16: Decomposition of Matrices
Any square matrix can be uniquely expressed as the sum of symmetric and skew-symmetric parts.
Formula: A = (A + A^T)/2 + (A - A^T)/2.
Page 17: Orthogonal Matrices
Orthogonal condition: A^TA = I, where I is an identity matrix.
Involves unit vectors that are mutually orthogonal.
Basic properties of unit vectors in relations.
Page 18: Cofactor and Adjoint
Adjoint of matrix A: Adj(A) = (Cofactor matrix of A)^T.
Finding the adjoint of matrix through cofactor expansion.
Page 19: Properties of Adjoint
Relation of adjoint to matrix A: A (adj(A)) = (det A) * I.
Additional properties of adjoint matrices concerning commutation and symmetry.
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Page 22: Cayley-Hamilton Theorem
Statement: Every square matrix satisfies its characteristic equation.
Characteristic polynomial det(A - XIn) = 0 leads to eigenvalues.
Application of the theorem in matrices.
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Page 24: Gauss Jordan Method
Solving system of linear equations using matrices.
Steps for both non-singular and singular matrices to determine solutions.
Application of adjoint and confirmation of null matrices for interpretation.