Linear Algebra (Part- II) Study Notes for GATE CSE

Inverse of a Matrix

Let us suppose a square matrix A, and its inverse is written A^-1. If we multiply A with A^-1 so the result will be an identity matrix I. Non-square matrices do not have inverses.

Note: Not all square matrices have inverses. A square matrix that has an inverse is known as nonsingular or invertible, and a square matrix without an inverse is known as singular or noninvertible.

Rank of a Matrix

The maximum number of linearly independent rows in matrix A is called the row rank of A, and the maximum number of linearly independent columns in A is called the column rank of A. If A is an m×n matrix, i.e., if A has m rows and n columns, so

1. row rank of A <= m
2. Column rank of A <= n

Eigen values and Eigenvectors

Let, A = [a_ij] is a square matrix of order n.  if there exists a non-zero column vector X and a scalar l such that AX = lX, then l is called the Eigenvalue of the matrix A and X is called an Eigenvector corresponding to the eigenvalue l.

Characteristic Matrix:

The matrix A – lI is called the characteristic matrix of given matrix A which is obtained by subtracting l from diagonal elements of A.

Characteristic Polynomial:

The determinant IA – lII when expanded, will give a polynomial of degree n in l which is called the characteristic polynomial of matrix A.

Characteristic equation:

The equation IA – lII = 0 is called the characteristic equation or secular equation of matrix A.

Characteristic Roots or Eigenvalues or Latent roots:

The roots of the characteristic eq.  l₁, l₂ ……… l_n are called characteristic roots or Eigenvalues or Latent roots.

Characteristic Vectors or Eigen Vectors:

Corresponding to each characteristic root l there corresponds non-zero vector X satisfying the equation (A – lI) X = 0.  The non-zero vectors X are called characteristic vectors or Eigenvectors.

The spectrum of a Matrix

The set of all Eigenvalues of given matrix A is called the spectrum of A

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