Posts Tagged ‘matrix’
Friday, November 6th, 2009
Identity Matrix Explained
Description
A detailed tutorial on the identity matrix. Step by step tutorial including several examples of the identity matrix and how to solve it for reference.
Overview
An indentity matrix is a matrix that is said to be of size n. It is considered to be the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. The identity matrix is denoted as the variable I. The identity matrix has some extremely important properties of its own, especially multiplication properties. It is a unique type of matrix that is found rarely, but is used very often in several different branches of math.
Tags: -1, 0, algebra, diagonal, i, identity, linear, main, matrices, matrix, multiplication, one, properties, square, uniquem, variable, zero
Posted in Algebra | No Comments »
Friday, November 6th, 2009
Introduction to Invariants
Description
A detailed tutorial on invariants and the property of invariance. Step by step tutorial including several examples of invariants for reference.
Overview
Invariants are any function or number that displays the property of invariance. Invariance is when a function or number can go through several transformations without changing, or without going outside of its set parameters. The set parameters differ depending on the function or number. Some examples of invariant functions and numbers are the absolute value of a complex number, the degree of a polynomial, and certain parts of a square matrix
Tags: absolute, arithmetic, complex, degree, determinant, eigenvalue, eigenvector, function, invariance, invariant, matrix, number, parameters, polynomial, square, trace, transformations, value
Posted in Arithmetic | No Comments »
Friday, November 6th, 2009
Introduction to Scalars
Description
A detailed tutorial on what a scalar is. Step by step tutorial including several examples of scalars and how they relate to vectors for reference.
Overview
A scalar is a number that relates vectors on a vector space through the process of scalar multiplication. A scalar can be taken from any set of numbers, including rational, algebraic, real, and complex sets of numbers. The scalar is always a real number. A scalar is a single component, and things such as vectors, matrices, and tensors can be reduced to a scalar.
Tags: algebra, algebraic, complex, component, compound, matrices, matrix, number, quaternions, rational, real, scalar, single, space, tensor, vector
Posted in Algebra | No Comments »
Thursday, November 5th, 2009
Main Diagonal of a Matrix
Description
A detailed tutorial on the main diagonal of a matrix. Step by step tutorial including several examples of main diagonals for reference.
Overview
The main diagonal of a matrix is the diagonal that starts at the top left corner, and continues down and to the right one step until either the other corner is reached (square matrices only), the bottom of the matrix is reached, or the right side of the matrix is reached. The main diagonal is also sometimes called the primary diagonal or the leading diagonal
Tags: algebra, bottom, diagonal, leading, left, linear, main, matrices, matrix, primary, regular, right, square, step, top
Posted in Algebra | No Comments »
Thursday, November 5th, 2009
Introduction to Linear Transformations
Description
A detailed tutorial on linear transformations. Step by step tutorial including several examples of linear transformations for reference.
Overview
A linear transformation takes place between two vector spaces. For two vector spaces V and W, there is a map T such that T(v_1 + v_2) = T(v_1) + T(v_2) for any vectors v_1 and v_2 in V, and T(a v) = a T(v) for any scalar a. Examples of linear transformation are often obtained through matrix multiplication. Linear transformations can also be injective or surjective
Tags: algebra, injective, linear, map, matrix, multiplication, scalar, space, surjective, transformation, vector
Posted in Algebra | No Comments »
Thursday, November 5th, 2009
Upper and Lower Triangular Matrices
Description
A detailed tutorial on upper and lower triangular matrices. Step by step tutorial including several examples of triangular matrices for reference.
Overview
A triangular matrix is a kind of square matrix where an element above or below the main diagonal is 0. This gives the true elements of the matrix a triangle shape, which is how it got its name. An upper triangular matrix is sometimes called a right triangular matrix. The matrix is up in the right upper corner, and the 0 element is in the lower left corner. A lower triangular matrix is sometimes called a left triangular matrix. The matrix is in the left bottom corner, and the 0 element is in the upper right corner.
Tags: 0, algebra, bottom, element, left, lower, matrices, matrix, right, square, top, triangle, triangular, upper, zero
Posted in Algebra | No Comments »
Thursday, November 5th, 2009
Transpose of a Matrix Explained
Description
A detailed tutorial on the transpose of a matrix. Step by step tutorial including several examples of the transpose of a matrix for reference.
Overview
When you transpose a matrix, it is simply a way of saying that you write the matrix in a different way – this creates a new matrix. There are three ways you can transpose a matrix. The first way is to write the rows of your matrix as columns instead. The second way is to write the columns of your matrix as rows instead. And the third way is to reflect your matrix by its main diagonal. All of these actions accomplish the same thing, so it does not matter which method you use. When people talk about transposing something, they are usually referring to matrices.
Tags: algebra, columns, diagonal, element, equivalent, main, matrices, matrix, method, reflect, rows, scalar, transpose
Posted in Algebra | No Comments »
Tuesday, November 3rd, 2009
Eigenvalues and Eigenvectors Explained
Description
A detailed tutorial on eigenvalues and eigenvectors. Step by step tutorial including several examples of eigenvalues and eigenvectors for reference.
Overview
Eigenvalues and eigenvectors are related concepts commonly used in linear algebra. More specifically, they are properties of a matrix. They give very important information about a matrix, and are used in matrix factorization. Assuming that a matrix is a diagonal matrix (a square matrix or a similar matrix that you can calculate diagonals on), then the eigenvalues are the numbers on the diagonal and the eigenvectors are the basis vectors to which there numbers refer. You cannot have an eigenvector without an eigenvalue. However, you can have an eigenvalue without an eigenvector.
Tags: algebra, basis, diagonal, eigenvalue, eigenvector, factorization, linear, matrices, matrix, properties, square, transformations, vectors
Posted in Algebra | No Comments »
Tuesday, November 3rd, 2009
Introduction to Square Matrices
Description
A detailed tutorial on square matrices and how to identify them. Step by step tutorial including several examples of square matrices for reference.
Overview
A square matrix is a simple matrix in the shape of a square. It has the same number of rows and columns. Square matrices are called nxn matrces. The most common values for n are 2 and 3. Two columns and rows is the smallest amount of rows and columns a square matrix can have – matrices with only one value are not considered to be square.
Tags: 2, 2x2, 3, 3x3, algebra, columns, equal, equivalent, linear, matrices, matrix, n!, number, nxn, rows, same, shape, square, three, two, values
Posted in Algebra | No Comments »
Tuesday, November 3rd, 2009
How to Find the Trace
Description
A detailed tutorial on find the trace of a matrix. Step by step tutorial including several examples of how to find the trace for reference.
Overview
The trace of a square matrix is defined to be the sum of the elements on the main diagonal of the matrix. This can be mathematically expressed as:
Remember, the trace is only defined for square matrices – not any other kind of matrix.
Tags: algebra, diagonal, eigenvalue, element, invariant, linear, main, matrices, matrix, Spur, square, sum, trace
Posted in Algebra | No Comments »
