Linear Algebra - II Study Notes for Civil Engineering

By Sachin Singh|Updated : November 14th, 2017

1. Inverse of a Matrix

For a square matrix A, the inverse is written A-1When A is multiplied by A-1 the result is the identity matrix I. Non-square matrices do not have inverses.

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

 

AA-1 = A-1A = I

Example:For matrix byjusexamprep, its inverse is byjusexamprep since
 


AA-1 = byjusexamprep

and A-1A = byjusexamprep.

 

Here are One ways to find the inverse of a matrix:

1. Shortcut for 2x2 matrices

For byjusexamprep, the inverse can be found using this formula:

byjusexamprep

Examplebyjusexamprep

2. Rank of a Matrix

The maximum number of linearly independent rows in a 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 by n matrix, that is, if A has m rows and n columns, then it is obvious that

  byjusexamprep

Example 1: Find the rank of the matrix

  byjusexamprep

First, because the matrix is 4 x 3, its rank can be no greater than 3. Therefore, at least one of the four rows will become a row of zeros. Perform the following row operations:

  byjusexamprep

Since there are 3 nonzero rows remaining in this echelon form of Bbyjusexamprep 

3. Eigenvalues and Eigenvectors

A = [aij] be 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 Eigen value of the matrix A and X is called an Eigen vector corresponding to the eigen value l.

Characteristic Matrix:

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

Characteristic Polynomial:

            The determinant |A – I| when expanded will give a polynomial of degree n in l which is called characteristic polynomial of matrix A.

Characteristic equation:

            The equation |A - II| = 0 is called characteristic equation or secular equation of matrix A.

Characteristic Roots or Eigenvalues or Latent roots:

            The roots of the characteristic eq.  l1, l2 ……… ln 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 - I). X = 0.  The non-zero vectors X are called characteristic vectors or Eigen vectors.

Spectrum of a Matrix:

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

 

Example: Find Eigenvalues and Eigenvectors of a 2x2 Matrix

If

 byjusexamprep

then the characteristic equation is 

 byjusexamprep

and the two eigenvalues are

λ1=-1, λ2=-2

All that's left is to find the two eigenvectors.  Let's find the eigenvector, v1, associated with the eigenvalue, λ1=-1, first.

byjusexamprep

so clearly from the top row of the equations, we get

byjusexamprep

Note that if we took the second row we would get

byjusexamprep 

In either case we find that the first eigenvector is any 2 element column vector in which the two elements have equal magnitude and opposite sign.

byjusexamprep

where k1 is an arbitrary constant.  Note that we didn't have to use +1 and -1, we could have used any two quantities of equal magnitude and opposite sign.

Going through the same procedure for the second eigenvalue:

byjusexamprep

 Again, the choice of +1 and -2 for the eigenvector was arbitrary;

 Thanks,

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