By Kardi Teknomo, PhD .
LinearAlgebra

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A matrix usually consists of many scalar elements. Can we characterize a matrix by a few numbers? In particular, our question is given a square matrix Eigen Value and Eigen Vector , can we find a scalar number Eigen Value and Eigen Vector and a vector Eigen Value and Eigen Vector such that Eigen Value and Eigen Vector ? Any solution of equation Eigen Value and Eigen Vector for Eigen Value and Eigen Vector is called eigenvector of Eigen Value and Eigen Vector . The scalar is called the eigenvalue of matrix Eigen Value and Eigen Vector .

Eigenvalue is also called proper value, characteristic value, latent value, or latent root. Similarly, eigenvector is also called proper vector, characteristic vector, or latent vector.

In the topic of Linear Transformation , we learned that a multiplication of a matrix with a vector will produce the transformation of the vector Eigen Value and Eigen Vector . Notice the equation Eigen Value and Eigen Vector said that multiplication of a matrix by a vector is equal to multiplication of a scalar by the same vector. Thus, the scalar Eigen Value and Eigen Vector characterizes the matrix Eigen Value and Eigen Vector .

Since eigenvalue Eigen Value and Eigen Vector is the scalar multiple to eigenvector Eigen Value and Eigen Vector , geometrically, eigenvalue indicates how much the eigenvector Eigen Value and Eigen Vector is shortened or lengthened after multiplication by the matrix Eigen Value and Eigen VectorEigen Value and Eigen Vector without changing the vector orientation .

Algebraically, we can solve the equation Eigen Value and Eigen Vector by rearranging it into a homogeneous linear system Eigen Value and Eigen Vector where matrix Eigen Value and Eigen Vector is the identity matrix order Eigen Value and Eigen Vector . A homogeneous linear system has non trivia solution if the matrix Eigen Value and Eigen Vector is singular . That happens when the determinant is equal to zero, that is Eigen Value and Eigen Vector . Equation is called the characteristic equation of matrix Eigen Value and Eigen Vector .

Expanding the determinant formula (using cofactor), we will get the solution in the polynomial form with coefficients Eigen Value and Eigen Vector . This polynomial equation Eigen Value and Eigen Vector is called the characteristic polynomial of matrix Eigen Value and Eigen Vector . The solution of the characteristic polynomial of Eigen Value and Eigen Vector are Eigen Value and Eigen Vector eigenvalues, some eigenvalues may be identical (the same eigenvalues) and some eigenvalues may be complex numbers.

Each eigenvalue has a corresponding eigenvector. To find the eigenvector, we put back the eigenvalue into equation Eigen Value and Eigen Vector . We do that for each of the eigenvalue. If Eigen Value and Eigen Vector is an eigenvector of A, then any scalar multiple is also an eigenvector with the same eigenvalue. We often use normalized eigenvector into unit vector such that the inner product with itself is one Eigen Value and Eigen Vector .

Example:
Find eigenvalues and eigenvectors of matrix Eigen Value and Eigen Vector
Solution: we form characteristic equation Eigen Value and Eigen Vector
Eigen Value and Eigen Vector
The eigenvalues are Eigen Value and Eigen Vector and Eigen Value and Eigen Vector .
For the first eigenvalue Eigen Value and Eigen Vector , the system equation is Eigen Value and Eigen Vector
Eigen Value and Eigen Vector
Eigen Value and Eigen Vector
The two rows are equivalent and produces equation Eigen Value and Eigen Vector . This is an equation of a line with many solutions, we can put arbitrary value Eigen Value and Eigen Vector to obtain
Eigen Value and Eigen Vector . You can also write as Eigen Value and Eigen Vector or Eigen Value and Eigen Vector and they lie on the same line.
The normalized eigenvector is Eigen Value and Eigen Vector
For the second eigenvalue Eigen Value and Eigen Vector , the eigenvector is computed from the system equation Eigen Value and Eigen Vector
Eigen Value and Eigen Vector
Eigen Value and Eigen Vector
The two rows are equivalent and produces equation Eigen Value and Eigen Vector . This is an equation of a line with many solutions, arbitrarily we can put Eigen Value and Eigen Vector to obtain
Eigen Value and Eigen Vector . You can also write as Eigen Value and Eigen Vector or Eigen Value and Eigen Vector and they lie on the same line.
The normalized eigenvector is Eigen Value and Eigen Vector















Thus, eigenvalue Eigen Value and Eigen Vector has corresponding eigenvector Eigen Value and Eigen Vector and eigenvalue Eigen Value and Eigen Vector has corresponding eigenvector Eigen Value and Eigen Vector . Note that the eigenvectors are actually lines with many solutions and we put only one of the solutions. They are correct up to a scalar multiple. Since the eigenvalues are all distinct, the matrix is diagonalizable and the eigenvectors are linearly independent .

Properties

Some important properties of eigenvalue , eigenvectors and characteristic equation are:

  • Every square matrix has at least one eigenvalue a corresponding non zero eigenvector.
  • When a square matrix has multiple eigenvalues (that is repeated, non-distinct eigenvalues), we have two terms to characterize the complexity of the matrix:
    • The algebraic multiplicity of an eigenvalue Eigen Value and Eigen Vector is the integer Eigen Value and Eigen Vector associated with Eigen Value and Eigen Vector when it appears in the characteristic polynomial. If the algebraic multiplicity is one, the eigenvalues is said to be simple .
    • The geometric multiplicity of Eigen Value and Eigen Vector is the number of linearly independent eigenvectors that can be associated with Eigen Value and Eigen Vector . For any eigenvalue, the geometric multiplicity is always at least one. Geometric multiplicity never exceeds algebraic multiplicity.
    Example:
    Matrix Eigen Value and Eigen Vector has characteristic polynomial Eigen Value and Eigen Vector thus the eigenvalue is 6 with algebraic multiplicity of 2. There is only one linearly independent eigenvector Eigen Value and Eigen Vector , thus the geometric multiplicity is 1.


  • The eigenvectors that belong to distinct eigenvalues are linearly independent eigenvectors. This is true even if the eigenvalues are not all distinct.
  • If a square matrix Eigen Value and Eigen Vector has fewer than Eigen Value and Eigen Vector linearly independent eigenvector, then matrix Eigen Value and Eigen Vector is called defective matrix . Defective matrix is not diagonalizable.
  • When the eigenvalues of a square matrix Eigen Value and Eigen Vector are all distinct (no multiple eigenvalues), we called it non-defective matrix. A non-defective matrix has Eigen Value and Eigen Vector linearly independent eigenvectors that can form a basis (i.e. a coordinate system) for Eigen Value and Eigen Vector dimensional space. Non-defective matrix Eigen Value and Eigen Vector is diagonalizable by similarity transformation Eigen Value and Eigen Vector into a diagonal matrix Eigen Value and Eigen Vector . The eigenvalues of Eigen Value and Eigen Vector lie on the main diagonal of Eigen Value and Eigen Vector . Modal matrix Eigen Value and Eigen Vector is formed by horizontal concatenation of the Eigen Value and Eigen Vector linearly independent eigenvectors Eigen Value and Eigen Vector .
  • If matrix Eigen Value and Eigen Vector is symmetric then matrix Eigen Value and Eigen Vector has linearly independent Eigen vectors and the Eigen values of symmetric matrix Eigen Value and Eigen Vector are all real numbers (no complex numbers).
  • If all eigenvalues of symmetric matrix Eigen Value and Eigen Vector are distinct (all eigenvalues are simple), then matrix Eigen Value and Eigen Vector can be transformed into a diagonal matrix. Furthermore, the eigenvectors are orthogonal.
  • Matrix Eigen Value and Eigen Vector satisfies its own characteristic equation . If polynomial Eigen Value and Eigen Vector is the characteristics polynomial equation of a square matrix A, then matrix Eigen Value and Eigen Vector satisfies Cayley-Hamilton equation Eigen Value and Eigen Vector .