Linear Algebra a new Isomorphism: The Matrix
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Number of Eigenvalues An n x n matrix has--at most--n real eigenvalues, even if they are counted with their algebraic multiplicities. If n is odd, there exists at least one real eigenvalue If n is even, there need not exist any real eigenvalues. Eigenvalues, the Determinant, and the Trace If an n x n matrix A has eigenvalues  listed with their algebraic multiplicities, then
Special Case: Triangular Matrix The eigenvalues of a Triangular Matrix are its diagonal entries. Special Case: Eigenvalues of Similar Matrices If matrix A is similar to matrix B (i.e. there exists an invertible S Э B = S-1AS): A & B have the same characteristic polynomial rank(A) = rank (B), nullity (A) = nullity (B) A and B have the same eigenvalues with the same algebraic and geometric multiplicities. Eigenvectors may be different! Matrices A and B have the same determinant and trace. Finding the Eigenvectors of a Matrix Eigenspace- For a particular eigenvalue of a matrix A, the kernel of the matrix  or 
The eigenvectors with eigenvalue are the nonzero vectors in the eigenspace. Geometric Multiplicity- the dimension of the eigenspace (the nullity of the matrix ).
If is an eigenvalue of a square matrix A, then the geometric multiplicity of must be less than of equal to the algebraic multiplicity of . Eigenbasis- a basis of Rn consisting of eigenvectors of A for a given n x n matrix A. If an n x n matrix A has n distinct eigenvalues, then there exists an eigenbasis for A. Eigenbasis and Geometric Multiplicities By finding the basis of every eigenspace of a given n x n matrix A and concatenating them, we can obtain a list of linearly independent eigenvectors (the largest number possible); if the number of elements in this list is equal to n (i.e. the geometric multiplicities sum to n), then we can construct an eigenbasis; otherwise, there doesn’t exist an eigenbasis. Diagonalization The process of constructing the matrix of a linear transformation with respect to the eigenbasis of the original matrix of transformation; this always produces a diagonal matrix with the diagonal entries being the transformation’s eigenvalues (recall that eigenvalues are independent of basis; see eigenvalues of similar matrices). 
Diagonalizable Matrix- An n x n matrix A that is similar to some diagonal matrix D. A matrix A is diagonalizable ∃ an eigenbasis for A If an n x n matrix A has n distinct eigenvalues, then A is diagonalizable. Powers of a Diagonalizable Matrix To compute the powers At of a diagonalizable matrix (where t is a positive integer), diagonalize A, then raise the diagonal matrix to the t power: 
Complex Eigenvalues Polar Form 
De Moivre’s Formula 
Fundamental Theorem of Algebra Any polynomial of degree n has, allowing complex numbers and algebraic multiplicity, exactly n (not necessarily distinct) roots. Finding Complex Eigenvectors After finding the complex eigenvalues, subtract them along the main diagonal like usual. Afterwards, however, simple take the top row of the resulting matrix, reverse it, multiply by a negative, and voila an eigenvector! This may only work for 2 x 2 matrices. Discrete Dynamical Systems Many relations can be represented as a ‘dynamical system’, where the state of the system is given, at any time, by the equation  or equivalently (by exponentiation)  where represents the basis state and  represents the state at any particular time, t.
This can be further simplified by first finding a basis  of Rn such that are eigenvectors of the matrix A. Then, using  , the exponentiated matrix At can be distributed and the eigenvector properties of utilized to generate Download 265.79 Kb. Share with your friends:
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