Introduction to Matrices

Introduction to Matrices

• Matrices are arrangements of elements in rows and columns.
• The lecture aims to cover matrices comprehensively.

Basic Definitions & Order

• A matrix is an arrangement of elements in rows and columns.
• An M x N matrix has M rows and N columns.
• "M" is the number of rows, representing horizontal arrangements.
• "N" is the number of columns, representing vertical arrangements.
• Order of a matrix is rows x columns (M x N).
• An M x N matrix contains M * N elements.
• Elements within a matrix are called entries or elements.
• Matrices can be real or complex based on their entries.

Representing Matrix Elements

• A11 represents the element in the first row, first column.
• A12 represents the element in the first row, second column.
• Aij represents the element in the i-th row, j-th column.
• In Aij, "I" is the row number and "J" is the column number.
• A matrix with M rows and N Columns can be represented as Aij (M x N).

Types of Matrices

• Square Matrix: Number of rows equals the number of columns (M = N).
• Rectangular Matrix: Number of rows doesn't equal number of columns (M != N).
• Null Matrix: A matrix with all entries as zero, indicated with "0".
• Horizontal Matrix: A matrix with more columns than rows, appearing wide.
• Vertical Matrix: A matrix with more rows than columns, appearing tall.

Square Matrix Detail

• An N x N matrix has N² elements.
• Diagonal elements are A11, A22, …, Ann; there are N diagonal elements.
• Elements above the diagonal have a lower row number (I < K). There are n²-n/2 elements.
• Elements below the diagonal have a higher row number (I > K). There are n²-n/2 elements.

Triangular Matrices

• Focus is on the positioning of zero-entries in Upper Triangular vs Lower Triangular matrices.

Upper Triangular Matrix

• Entries below the diagonal are all zero.
• Aij = 0 for all i > j.

Lower Triangular Matrix

• Entries above the diagonal are all zero.
• Aij = 0 for all i < j.

Diagonal Matrix

• Non-diagonal entries are zero.
• Diagonal entries can be any value (zero or non-zero).

Scalar Matrix

• A diagonal matrix where all diagonal entries are equal.

Identity/Unit Matrix

• A diagonal matrix where all diagonal entries are 1; denoted by "I".
• The number of rows and columns must be specified (e.g., 3x3).

Minimum Zeroes in Special Matrices

• Upper/Lower Triangular: Minimum number of zeroes = (n² - n) / 2.
• Diagonal/Scalar/Unit Matrix: Minimum number of zeroes = n² - n.

Matrix vs Determinant

• A matrix is an arrangement without a specific value.
• A determinant has a numerical value calculated from a square matrix.
• Every square matrix can have an associated determinant with elements in the same order.

Singular and Non-Singular

• A singular matrix has a determinant of zero.
• A non-singular matrix has a determinant that is not zero.
• A zero matrix, if square, has a determinant of zero: Det(0)=0.

Determinant of Special Matrices

• Determinant of an Upper Triangular, Lower Triangular, or Diagonal matrix is the product of its diagonal entries.
• The determinant of an identity matrix is always 1.
• If the null matrix is of order n, its determinant is zero

Algebra of Matrices: Equality

• Matrices are equal only if their order is the same and corresponding elements are equal.

Algebra of Matrices: Edition

• Sum of matrices A+B involves summing corresponding elements.
• Iij=Bji
• Addition is possible only for matrices of the same order.
• The additive inverse for any matrix A is its negative version B, where A + B = 0. Thus B is A's inverse.
• Scalar multiplication involves multiplying every element in matrix A by a constant “k”.

Multiplication of Matrices

• Number of Columns in A must equal the number of rows in B.
• The new matrix will have the same number of Rows as A and Columns as B.
• If Order of A is m x n, and Order of B is n x p, then the order of the product is m x p.
• Multiplication is done Row by Column (A)x(B).
• Number of columns of A must equal the number of Rows in order for Multiplication to be valid.
• Multiplication involves Row by Column operations such as R1C1+R1C2+R1C…

Multiplication Properties

• AB=0 does not necessarily mean A=0 or B=0; two non-zero matrices can result in a zero matrix after multiplication.
• Matrix multiplication is associative: A(BC) = (AB)C.
• Matrix multiplication is distributive over addition.

Commutative and Anti-Commutative

• Matrices A and B commute if AB = BA.
• Matrices A and B anti-commute if AB = -BA.

Matrix Trace

• Matrix Trace only works with Squared Matrices.
• It is the SUM of all the diagonal Elements from a Matrix
• Represented using Sigma notation.
• Trace(KA}=K Trace(A)

Determinant vs Constant Multiplier

• The determinant with “n” can be exemplified by a 3rd order matrix, so 2 with the determinant is 2^3 to perform the operation.
• The determinant of a matrix with “K” to power “n” due to “n” being the number of times it repeats on the Matrix.

Properties of Trace

• Tr(A+B)=Tr(A=TR(B)
• Tr(lambda) = lambda tr(A)
• Tr(AB)= Tr(BA)

Transpose of the Matrix

• Changing Rows by Columns and Vice versa
• If the matrix is (mn) the new one would be (nm)
• The transpose of doing twice it would be the original
• (A+B)t = At +Bt, The sum of two transpose matrices is the sum of the original transposes.
• (KA)t=K(At), Multiplier (constant)
• (AB)t = (Bt)(At), Reversal Law of Transpose

Symmetric and Skew Matrices

• Squared Matrices
• Transpose matrix to be equal to the original is SIMETRIC, or at =a
• Symmetric Matrices have, at most, (n*n+1)/2 distinct entries.

Skew Matrix

• aT = -a, all the diagonals are “0” to be skew
• Determinant value of a Skew Symmetric Matrix with odd order = 0
• If A is symmetric and skew, then the result is the null Matrix.

The power of symmetric matrix properties

• If A is symmetric, then K(A), A(transpose), and An are also symmetric.
• If “n” is even, it's symmetrically skewed. If “n” is odd, it remains skewed.

The simetric matrix is equal to ½ (a)=A(transpose)

• All the square ones can be expressed by simetric Matrix
• A= 1/2[(A+AT)+(A-AT)]

Matrix Properties: Powers

• Only squared matrices can have powers, ex A^2
• A^3 = AAA = A^2(A) = A(A^2)
• Law of Exponents
• If AB=BA (binomial expansion) if not we need to solve for the whole formula

Nilportent Matrices: Power must be Identities matrices

• Power is only applicable with squared matrices.
• Nilpotent of Index 3 is raised to the power of 2.
• They can be squared with the identity or CAN'T be.
• Determinant Value of a Nilpotent Matrix must be Zero.

INVOLUNTARY matrix: Squared power equal to identities

• Values are either -1 or 1, never similar though.
• Value is similar but cannot be similar
• Value os 1 and or -1
• Matrix self inversed

Matrix periodic

• Square (K)=(K+N)= “A” must have a positive integer and is called Periodic.
• Period of identities is ONE

Orthogonal matrix

• If Multiplied by Transpose is Identity A At= I- Adjoint Matrix
• (AX)=C where (A)= coefficient, (x)= variable with unknown, (B)= constant matrix

Charactristic equation

• A squared matrix, ( a lambada equation) to form a polynomial equation, hence squared minus lambda
• A * AX= A, so AX-AB=0, and solved by A(transpose)=0

Keye Hamilton

• Every square matrix satisfies its own characteristic equation
• Allows finding inverse matrix.
• Allows simplifying higher powers of a matrix.