Linear Algebra

Inverse by Partitioning

• Inverse of a matrix can be calculated by partitioning the matrix into smaller sub-matrices
• This method is useful for finding the inverse of a large matrix
• Partitioning involves dividing the matrix into four sub-matrices and then finding the inverse of each sub-matrix

Rank of a Matrix

• Rank of a matrix is the maximum number of linearly independent rows or columns in the matrix
• It is a measure of the dimension of the range of the matrix
• Rank is used to determine the solvability of a system of linear equations

Rank-Nullity Theorem

• The rank-nullity theorem states that the rank of a matrix plus the nullity of the matrix is equal to the number of columns of the matrix
• This theorem is used to determine the dimension of the null space of a matrix
• The nullity of a matrix is the number of linearly independent vectors in the null space of the matrix

System of Linear Equations

• A system of linear equations is a set of equations where each equation is in the form of ax + by = c
• The solution to a system of linear equations is the set of values that satisfy all the equations
• The rank of the coefficient matrix determines the solvability of the system

Eigen Values and Eigen Vectors

• Eigen values are scalars that represent how much the matrix stretches or compresses a vector
• Eigen vectors are non-zero vectors that, when transformed by the matrix, result in a scaled version of themselves
• Eigen values and eigen vectors are used to diagonalize a matrix and to understand the structure of the matrix

Cayley-Hamilton Theorem

• The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation
• This theorem is used to find the eigen values of a matrix and to diagonalize a matrix
• The characteristic equation is a polynomial equation in which the matrix is a root