{"id":5125,"date":"2021-08-31T11:24:42","date_gmt":"2021-08-31T05:54:42","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=5125"},"modified":"2026-02-06T05:26:53","modified_gmt":"2026-02-05T23:56:53","slug":"what-is-the-characteristic-equation","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-the-characteristic-equation\/","title":{"rendered":"What is the characteristic equation?"},"content":{"rendered":"\n

Overview- characteristic equation<\/h2>\n\n\n\n

The characteristic equation is the equation which is used to find the Eigenvalues of a matrix.<\/p>\n\n\n\n

This is also called the characteristic polynomial.<\/p>\n\n\n\n

Definition- <\/strong>Let A be a square matrix, be any scalar then  is called the characteristic equation of a matrix A.<\/p>\n\n\n\n

Note:<\/p>\n\n\n\n

Let a be a square matrix and \u2018 \u2019 be any scalar then,<\/p>\n\n\n\n

1)     is called characteristic matrix<\/p>\n\n\n\n

2)     is called characteristic polynomial.<\/p>\n\n\n\n

The roots of a characteristic equation are known as characteristic root or latent roots, Eigenvalues<\/a> or proper values of a matrix A.<\/p>\n\n\n\n

Eigen vector<\/strong><\/h2>\n\n\n\n

Suppose be an Eigenvalue<\/a> of a matrix A. Then for every a non \u2013 zero vector x1<\/sub> such that.<\/p>\n\n\n\n

                                                      \u2026 (1)<\/p>\n\n\n\n

Such a vector \u2018x1<\/sub>\u2019 is called an Eigenvector<\/a> corresponding to the Eigenvalue .<\/p>\n\n\n\n

Properties of Eigenvalues<\/strong><\/h2>\n\n\n\n

1.     The sum of the Eigenvalues of a matrix A is equal to the sum of the diagonal elements of a matrix A.<\/p>\n\n\n\n

2.     The product of all Eigenvalues of a matrix A is equal to the value of the determinant.<\/p>\n\n\n\n

3.     If are n Eigenvalues<\/a> of square matrix A then are m Eigenvalues of a matrix A-1<\/sup>.<\/p>\n\n\n\n

4.     The Eigenvalues of a symmetric matrix are all real.<\/p>\n\n\n\n

5.      If all Eigenvalues are non \u2013zero then A-1<\/sup> exist and conversely.<\/p>\n\n\n\n

6.     The Eigenvalues of A and A\u2019 are the same.<\/p>\n\n\n\n

Properties of Eigenvector<\/strong><\/h2>\n\n\n\n

1.     Eigenvectors corresponding to distinct Eigenvalues are linearly independent.<\/p>\n\n\n\n

2.     If two more Eigen values are identical then the corresponding Eigenvectors may or may not be linearly independent.<\/p>\n\n\n\n

3.     The Eigenvectors corresponding to distinct Eigenvalues of a real symmetric matrix are orthogonal.<\/p>\n\n\n\n

Example: Find out the Eigenvalues and Eigenvectors of <\/strong><\/strong><\/p>\n\n\n\n

Sol.  The Characteristics equation is given by <\/strong><\/strong><\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Hence the Eigen values are 0, 0 and 3.<\/p>\n\n\n\n

The Eigen vector corresponding to Eigen value is<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Where X is the column matrix of order 3 i.e. <\/p>\n\n\n\n

This implies that x + y + z = 0<\/p>\n\n\n\n

Here the number of unknowns is 3 and the number of equations is 1.<\/p>\n\n\n\n

Hence we have (3-1) = 2 linearly independent solutions.<\/p>\n\n\n\n

Let z = 0, y =  1 then x = -1 or let z = 1, y = 1 then x = -2<\/p>\n\n\n\n

Thus the Eigenvectors corresponding to the Eigenvalue are (-1,1,0) and (-2,1,1).<\/p>\n\n\n\n

The Eigenvector corresponding to Eigenvalue is<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Where X is the column matrix of order 3, it means <\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

This implies that -2x + y + z = 0<\/p>\n\n\n\n

x – 2y + z = 0<\/p>\n\n\n\n

x + y – 2z = 0<\/p>\n\n\n\n

Taking last two https:\/\/lk.pusk.mipt.ru\/<\/a> equations we get<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Or<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Thus the Eigenvectors corresponding to the Eigenvalue are (3,3,3).<\/p>\n\n\n\n

Hence the three Eigenvectors obtained are  (-1,1,0), (-2,1,1) and (3,3,3).<\/p>\n\n\n\n

Interested in learning about similar topics? Here are a few hand-picked blogs for you!<\/p>\n\n\n\n