**Prerequisites:**

- Mathematics | Eigen Values and Eigen Vectors
- Matrix Multiplication
- Null Space and Nullity of a Matrix

For a given matrix *A* the set of all eigenvectors of *A* associated with an eigenvalue **Eigenspace** of *A* with respect to *A* is called **Eigenspectrum**, or just spectrum, of *A*.

If

Below are some useful properties of eigenvalues and eigenvectors in addition to the properties which are already listed in the article Mathematics | Eigen Values and Eigen Vectors.

**Note:** ker stands for **Kernel** which is another name for *null space*.

**Computing Eigenvalues, Eigenvectors, and Eigenspaces:**

Consider given 2 X 2 matrix:Step 1: Characteristic polynomial and Eigenvalues.The characteristic polynomial is given by det() After we factorize the characteristic polynomial, we will get which gives eigenvalues as and Step 2: Eigenvectors and EigenspacesWe find the eigenvectors that correspond to these eigenvalues by looking at vectors x such thatFor we obtain After solving the above homogeneous system of equations, we will obtain a solution space This eigenspace is one dimensional as it possesses a single basis vector. Similarly, we find eigenvector for by solving the homogeneous system of equations This means any vector , where such as is an eigenvector with eigenvalue 2. This means eigenspace is given as

The two eigenspaces

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the **DSA Self Paced Course** at a student-friendly price and become industry ready.

## Recommended Posts:

- Mapping external values to dataframe values in Pandas
- Python | Visualize missing values (NaN) values using Missingno Library
- All possible values of floor(N/K) for all values of K
- Find trace of matrix formed by adding Row-major and Column-major order of same matrix
- Program to check diagonal matrix and scalar matrix
- Check if matrix can be converted to another matrix by transposing square sub-matrices
- Maximum trace possible for any sub-matrix of the given matrix
- Create matrix whose sum of diagonals in each sub matrix is even
- Construct a square Matrix whose parity of diagonal sum is same as size of matrix
- Minimize count of adjacent row swaps to convert given Matrix to a Lower Triangular Matrix
- Count right angled triangles in a matrix having two of its sides parallel to sides of the matrix
- Print Matrix after multiplying Matrix elements N times
- Construct a Matrix such that each cell consists of sum of adjacent elements of respective cells in given Matrix
- Find minimum possible values of A, B and C when two of the (A + B), (A + C) and (B + C) are given
- Comparing X^Y and Y^X for very large values of X and Y
- Number of subsets with same AND, OR and XOR values in an Array
- Calculate Bitwise OR of two integers from their given Bitwise AND and Bitwise XOR values
- Minimize array sum by replacing greater and smaller elements of pairs by half and double of their values respectively atmost K times
- Find smallest values of x and y such that ax - by = 0
- Find the minimum value of m that satisfies ax + by = m and all values after m also satisfy

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.