“They are common in many areas and quite important to solve problem.”

Eigenvalues and Eigenvectors of a matrix

In this section, we will work with the entire set of complex numbers, denoted by $\mathbb{C}$$\mathbb C$. Recall that the real numbers, $\mathbb{R}$$\mathbb R$ are contained in the complex numbers, so the discussions in this section apply to both real and complex numbers.

To illustrate the idea behind what will be discussed, consider the following example.

Compute $AX$$AX$ separately for

What do you notice about $AX$$AX$ in each of these products?

Notice that for each, $AX=kX$$AX = kX$ where $k$$k$ is some scalar. When this equation holds for some $X$$X$ and $k$$k$, we call the scalar $k$$k$ an eigenvalue of $A$$A$.

When $AX=kX$$AX = kX$ for some $X\ne 0$$X \neq 0$, we call such $X$$X$ an eigenvector of the matrix $A$$A$.

Definition 7.1.1: Eigenvalues and Eigenvectors

Let A be an n×n matrix and let X∈Cn

be a nonzero vector for which

for some scalar λ. Then λ is called an eigenvalue of the matrix A and X is called an eigenvector of A associated with λ, or a λ-eigenvector of A.

The set of all eigenvalues of an n×n

matrix A is denoted by σ(A) and is referred to as the spectrum of A.

Haven’t Handled

Matrix

Understand the concept of eigenvalues and eigenvectors of an matrix.

inner product (also called the dot product )

row rank , column rank and normal rank

Geometric Interpretation of Eigenvalues and Eigenvectors

Determinant and Eigenvalues

Trace and Eigenvalues

Nullspace