By Kardi Teknomo, PhD.
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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, can we find a scalar number and a vector such that? Any solution of equation for is called eigenvector of. The scalar is called the eigenvalue of matrix.
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. Notice the equation said that multiplication of a matrix by a vector is equal to multiplication of a scalar by the same vector. Thus, the scalar characterizes the matrix.
Since eigenvalue is the scalar multiple to eigenvector, geometrically, eigenvalue indicates how much the eigenvectoris shortened or lengthened after multiplication by the matrixwithout changing the vector orientation.
Algebraically, we can solve the equation by rearranging it into a homogeneous linear system where matrix is the identity matrix order. A homogeneous linear system has non trivia solution if the matrix is singular. That happens when the determinant is equal to zero, that is. Equation is called the characteristic equation of matrix.
Expanding the determinant formula (using cofactor), we will get the solution in the polynomial form with coefficients. This polynomial equation is called the characteristic polynomial of matrix. The solution of the characteristic polynomial of are 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. We do that for each of the eigenvalue. If 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.
Example:
Find eigenvalues and eigenvectors of matrix
Solution: we form characteristic equation
The eigenvalues are and.
For the first eigenvalue, the system equation is
The two rows are equivalent and produces equation. This is an equation of a line with many solutions, we can put arbitrary value to obtain
. You can also write as or and they lie on the same line.
The normalized eigenvector is
For the second eigenvalue, the eigenvector is computed from the system equation
The two rows are equivalent and produces equation. This is an equation of a line with many solutions, arbitrarily we can put to obtain
. You can also write as or and they lie on the same line.
The normalized eigenvector is
Thus, eigenvalue has corresponding eigenvector and eigenvalue has corresponding eigenvector. 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 is the integer associated with when it appears in the characteristic polynomial. If the algebraic multiplicity is one, the eigenvalues is said to be simple.
- The geometric multiplicity of is the number of linearly independent eigenvectors that can be associated with. For any eigenvalue, the geometric multiplicity is always at least one. Geometric multiplicity never exceeds algebraic multiplicity.
Example:
Matrix has characteristic polynomial thus the eigenvalue is 6 with algebraic multiplicity of 2. There is only one linearly independent eigenvector, 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 has fewer than linearly independent eigenvector, then matrix is called defective matrix. Defective matrix is not diagonalizable.
- When the eigenvalues of a square matrix are all distinct (no multiple eigenvalues), we called it non-defective matrix. A non-defective matrix has linearly independent eigenvectors that can form a basis (i.e. a coordinate system) for dimensional space. Non-defective matrix is diagonalizable by similarity transformation into a diagonal matrix . The eigenvalues of lie on the main diagonal of . Modal matrix is formed by horizontal concatenation of the linearly independent eigenvectors.
- If matrixis symmetric then matrix has linearly independent Eigen vectors and the Eigen values of symmetric matrix are all real numbers (no complex numbers).
- If all eigenvalues of symmetric matrix are distinct (all eigenvalues are simple), then matrix can be transformed into a diagonal matrix. Furthermore, the eigenvectors are orthogonal.
- Matrix satisfies its own characteristic equation. If polynomial is the characteristics polynomial equation of a square matrix A, then matrix satisfies Cayley-Hamilton equation.
See also: Matrix Eigen Value & Eigen Vector for Symmetric Matrix, Similarity and Matrix Diagonalization, Matrix Power
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Teknomo, Kardi (2011) Linear Algebra tutorial. http:\\people.revoledu.com\kardi\
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