Fundamental Theorem Algebra

In the previous section we have developed principal components analysis (PCA) from the fundamental theorem of singular value decomposition (SVD). In particular we have shown by means of eq. (31.1) how an nxp rectangular data matrix X can be decomposed into an nxr orthonormal matrix of row-latent vectors U, a pxr orthonormal matrix of column-latent vectors V and an rxr diagonal matrix of latent values A. Now we focus on the geometrical interpretation of this algebraic decomposition. 

Fundamental Theorem of Algebra Every polynomial of degree n has exactly n real or complex roots, counting multiplicities. 

We will need the Fundamental Theorem of Linear Algebra in Section 7.1. 

Proposition 2.5 (Fundamental Theorem of Linear Algebra) For any linear transformation T with finite-dimensional domain V we have... 

In Section 7.1 we will use this characterization of homogeneous harmonic polynomials as a kernel of a linear transformation (along with the Fundamental Theorem of Linear Algebra, Proposition 2.5) to calculate the dimensions of the spaces of the spherical harmonics. 

Proof. Consider the characteristic polynomial of T, that is, det (A/ — T). Because of the A" term, this complex-coefficient polynomial has degree n > 0. Hence, by the Fundamental Theorem of Algebra, this polynomial has at least one complex root. In other words, there exists a A e C such that det(A/ — T) = 0. This implies that there is a nonzero r> e V such that (A/ — r)u = 0. Hence Xv = Tv and A is an eigenvalue of T. ... 

This proof does not give a method for finding real eigenvalues of real linear operators, because the Fundamental Theorem of Algebra does not guarantee real roots for polynomials with real coefficients. Proposition 2.11 does not hold for inhnite-dimensional complex vector spaces eiffier. See Fxercise 2.28. 

For a proof of the Fundamental Theorem of Algebra, see any abstract algebra textbook,... 

To calculate the dimension of the vector space H for every nonnegative integer f we will use the Fundamental Theorem of Linear Algebra (Proposition 2.5), which we repeat here if T is a linear transformation from a finitedimensional vector space V to a finite-dimensional vector space W, then we have... 

Our goal is to calculate the dimension of the kernel of V, since this kernel consists precisely of the harmonic functions in. From Section 2.2 we know that the dimension of P is (f -I- l)(f -f 2). So, by the Fundamental Theorem of Linear Algebra (Proposition 2.5) it suffices to calculate the dimension of the image of the the linear transformation V. ... 

Some real matrix classes, studied in subsection (F) below, however, have only real eigenvalues and corresponding real eigenvectors. The complication with complex eigenvalues and eigenvectors is caused by the Fundamental Theorem of Algebra which states that all the roots of both real and complex polynomials can only be found in the complex plane C. 

Thus we have derived an algebraic equation of A -th degree, which has, in accordance with the fundamental theorem of algebra, N roots. However, sometimes it is not as important to calculate the roots as to analyse their properties, e.g. to find out how many positive, negative, zero roots the equation has, and so on. In such cases we have to find the coefficients of the characteristic polynomial. But how ... 

The fundamental theorem of algebra states that a polynomial of degree n has n number of roots, although some of the roots may be complex numbers. If a root of a polynomial of degree n is known to be, say r, then a polynomial of degree n - 1 is given by... 

Similarly, the number of linearly independent rows of A is called the row-rank of A. The row-rank of A is the column-rank of A. A fundamental theorem in matrix algebra states that the row-rank and the column-rank of a matrix are equal (and equal to the rank) [Schott 1997], Hence, it follows that the rank r(A) < min(/,/). The matrix A has full rank if and only if r(A) = mini/,/). Sometimes the term full column-rank is used. This means that r(A) = min(/,/) =. /, implying that J < I. The term full row-rank is defined analogously. 

Exercise 2.9 Null space and fundamental theorem of linear algebra... 

Equation 2.67 in Exercise 2.8 is begging to be analyzed by the fundamental theorem of linear algebra [7], so we explore that concept here. Consider an arbitrary m x n matrix, B. The null space of matrix B, written 3 B), is defined to be the set of all vectors x such that S x = 0. The dimension of SEiB) is the. number of linearly independent vectors x satisfying B x == 0. One of the remarkable results of the fundamental theorem of linear algebra is the relation... 

It tells us that if A, B, C are any three points of one Une and A, B, C are any three points of either the same or another line, a projective mapping may be constructed (as in 3.11(b)) in which the images of A,B,C are A, B, C respectively. This construction can be made in a great variety of ways but the fundamental theorem states that the actual mapping will always be the same. Irrespective of how the construction is made, the image of a fourth point D will be a luhque point D. Expressed algebraically this condition follows from the cross ratio ... 

Gauss Karl Friedrich (1777-1855) Ger. math., fundamental theorems of algebra and ontribution to modern number theory (Gaussian integers), vectorial representation of complex numbers, method of least squares and observational of errors, unit of magnetic field gauss named in his honor... 

It is concluded that the B(3) component in the field interpretation is nonzero in the light-like condition and in the rest frame. The B cyclic theorem is a Lorentz-invariant, and the product B x B<2> is an experimental observable [44], In this representation, B(3> is a phaseless and fundamental field spin, an intrinsic property of the field in the same way that J(3) is an intrinsic property of the photon. It is incorrect to infer from the Lie algebra (796) that Ii(3) must be zero for plane waves. For the latter, we have the particular choice (803) and the algebra (796) reduces to... 

The fundamental formula of the method of partial fractions is a theorem of algebra that says that if Q(x) is given by Eq. (5.45) and P x) is of lower degree than Q x), then... 

---