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The determinant of a square matrix is a number associated with the matrix, which is fundamental for the study of a square matrix; for example, a square matrix is invertible if and only if it has a nonzero determinant and the eigenvalues of a square matrix are the roots of a polynomial determinant.
In mathematics, particularly in linear algebra and applications, matrix analysis is the study of matrices and their algebraic properties. [1] Some particular topics out of many include; operations defined on matrices (such as matrix addition, matrix multiplication and operations derived from these), functions of matrices (such as matrix exponentiation and matrix logarithm, and even sines and ...
Several important classes of matrices are subsets of each other. This article lists some important classes of matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries. Matrices have a long history of both study and application, leading to ...
Any rotation matrix of size n × n can be constructed as a product of at most n(n − 1) / 2 such rotations. In the case of 3 × 3 matrices, three such rotations suffice; and by fixing the sequence we can thus describe all 3 × 3 rotation matrices (though not uniquely) in terms of the three angles used, often called Euler angles.
Matrix multiplication is defined in such a way that the product of two matrices is the matrix of the composition of the corresponding linear maps, and the product of a matrix and a column matrix is the column matrix representing the result of applying the represented linear map to the represented vector. It follows that the theory of finite ...
Similarity of matrices does not depend on the base field: if L is a field containing K as a subfield, and A and B are two matrices over K, then A and B are similar as matrices over K if and only if they are similar as matrices over L. This is so because the rational canonical form over K is also the rational canonical form over L. This means ...
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The above procedure shows why taking the pseudoinverse is not a continuous operation: if the original matrix has a singular value 0 (a diagonal entry of the matrix above), then modifying slightly may turn this zero into a tiny positive number, thereby affecting the pseudoinverse dramatically as we now have to take the ...