Find the characteristic polynomial $p(t)$ of $A$.Let $A$ be the $n\times n$ matrix that you want to diagonalize (if possible). A concrete example is provided below, and several exercise problems are presented at the end of the post. The process can be summarized as follows. We only describe the procedure of diagonalization, and no justification will be given. Here we explain how to diagonalize a matrix. A Hermitian Matrix can be diagonalized by a unitary matrix.
Step 5: Define the invertible matrix $S$.Step 4: Determine linearly independent eigenvectors.Step 1: Find the characteristic polynomial.The norm of a matrix may be thought of as its magnitude or length because it is a nonnegative number. For example, a trivial distance that has no equivalent norm is d( A, A) = 0 and d( A, B) = 1 if A ≠ B. However, not all distance functions have a corresponding norm. Once a norm is defined, it is the most natural way of measure distance between two matrices A and B as d( A, B) = ‖ A − B‖ = ‖ B − A‖. Since the set of all matrices admits the operation of multiplication in addition to the basic operation of addition (which is included in the definition of vector spaces), it is natural to require that matrix norm satisfies the special property: Triangle inequality: ‖ A + B‖ ≤ ‖ A‖ + ‖ B‖. Homogeneity: ‖ k A‖ = | k| ‖ A‖ for arbitrary scalar k. Is a function from a real or complex vector space to the nonnegative real numbers that satisfies the following conditions: In order to determine how close two matrices are, and in order to define the convergence of sequences of matrices, a special concept of matrix norm is employed, with notation \( \| \|. The set ℳ m,n of all m × n matrices under the field of either real or complex numbers is a vector space of dimension m Introduction to Linear Algebra with Mathematica Glossary Return to the main page for the second course APMA0340 Return to the main page for the first course APMA0330 Return to Mathematica tutorial for the second course APMA0340 Return to Mathematica tutorial for the first course APMA0330 Return to computing page for the second course APMA0340 Return to computing page for the first course APMA0330 Laplace equation in spherical coordinates.Numerical solutions of Laplace equation.Laplace equation in infinite semi-stripe.Boundary Value Problems for heat equation.Part VI: Partial Differential Equations.Part III: Non-linear Systems of Ordinary Differential Equations.Part II: Linear Systems of Ordinary Differential Equations.
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