and
[In 1960-1961, I had all of the pieces needed for the complex algorithms; but, lacking any motivation to do so, I did not develop them. In 1981, upon reading that "While there are many good algorithms for finding the eigenvalues/eigenvectors for a Hermitian matrix, there is no known algorithm for a [general] complex matrix.", I extended the foregoing algorithms to the complex domain. The hardware and software that was available to me then was inadequate. As a result, in the available time, the complex portion of the library was incomplete. Only now (in February 2004), I successfully implemented the capability to operate upon complex matrices.]
Why is it that textbooks on Linear Algebra eschew any discussion of complex matrices? I know not. The subject of complex matrices is just as reach as that of real matrices. The subject of complex matrices is no more difficult than that of real matrices. Insooth, the following theorem subsumes the study of the complex matrices into that of the real matrices.
Theorem: The complex-matrix C is isomorphic to the sum (A + i B), which in turn is isomorphic to the real-matrix (A, B; -B, A). The real-matrices A and B are obtained as the real and imaginary parts of the given complex-matrix C. This isomorphism is under:
transposition of a matrix (versus Hermitian transpose, by definition),
multiplication of a matrix by a real scalar, we provide a subroutine for the multiplication by a complex scalar.
matrix addition (hence, matrix subtraction),
matrix multiplication (hence, matrix inversion).
Proof: That between C and (A + i B) is obvious. Also obvious is that for addition between (A + i B) and (A, B; -B, A). We will prove only the multiplication between (A + i B) and (A, B; -B, A). Consider the product of these two matrices (A + i B) (C + i D) = ((A C - B D) + i (A D + A D). Multiply the corresponding two matrices (A, B; - B, A) (C, D; -D, C) = (A C - B D, A D + B C; A D + B C, A C - B D). QED.
Furthermore, the symmetries -- symmetric, skew-symmetric, or ortho-normal -- carry-over to the corresponding Hermitian, skew-Hermitian, or unitary. They are the same, except for the names. However, since the trace and determinant are not matrix-operations, it is not unexpected that they are not isomorphic. We provide a specific subroutine to compute the Hermitian trace. The determinant of the real-isomorph is the square of the magnitude of the determinant of the complex matrix. We do not provide for the computation of its phase-angle.
Once we have obtained the real version, all of the real definitions, theorems, and algorithms – as well as the whole real library – is at our disposal. We do not need any subroutines explicitly for complex matrices. Of this real library, only six routines required some enhancement to be able to handle the complex matrices.
The Frobenious matrix norm requires one line of code to multiply by half for the real isomorph of a complex matrix. The two sorting subroutines, the pseudo-random permutation matrix, and the Gramm-Schmidt ortho-normalization routines operate on the left-half of the complex matrix. Then each calls the same subroutine to cross-copy the left-half to the right-half. This customization is comprised of the same two lines of code in each of these four subroutines. The cross-copy subroutine consists of ten lines of code. This is a total of eighteen lines of code here and twenty-three more in the following Jacobi routine. The four routines that inter-convert between either the complex matrix or the (A + i B), and the (A, B; -B, A) isomorphs consist of a total of fifty lines of code, for a grand-total of ninety-two lines of code to adapt our real library to operate upon complex matrices. Also there are perhaps half-a-dozen routines specifically for complex matrices. However, they would be required, anyhow, as part of a complex library – if we had written a such explicitly. Hence, we obtain the equivalent of a complex library for only ninety-two lines of code!
The factorization of A = < D > is a matrix-multiplication; hence, it is isomorphic. The Jacobi eigenvalue-eigenvector (eigenpair) algorithm requires some additional capability: For a real matrix, each kill is an independent event. Since the complex matrix has two copies of each element, the kills have to be performed in complementary pairs, to maintain the Hermitian property of the matrix being factored and the unitary property of the bra being constructed. For the kill of a real element, perform the same kill upon each of the two copies of a given element, employing the same sin(x), cos(x), and delta. It is obvious that each of these requirements is satisfied thereby. For the kill of an imaginary element, we could attempt to maintain directly the Hermitian property of the matrix being factored. It would be tedious to perform this calculation by hand. Instead, we remember that matrices constitute a division ring. Hence, it suffices to maintain the unitary property of the bra being constructed. Then, the matrix being factored will take care of itself. Whenever we kill an imaginary element, we have to kill its symmetrically-positioned, about the imaginary diagonal, mate. To maintain the complex property of the bra being constructed, it is obvious that the same sin(x) and cos(x) have to be employed. The delta is given by delta = 2 b tan(x). Since the b – the value of the element being killed – changes sign, so must the delta.
Consider the converse. Now that we have implemented all of the theorems/algorithms/routines for the real isomorph of a complex matrix., what are the ramifications of going back to the complex matrix itself? An n by n real matrix requires 8 n^2 bytes of storage; a complex matrix 2 * (real); the real isomorph of a complex matrix 4 * (real). For today’s computers, these storage requirements are modest and the potential saving of going from the real isomorph to the corresponding complex matrix is insignificant. The computational requirements scale as n^2 or lower power for any routines not mentioned in the following discussion. Multiplication of a pair of real matrices requires n^3 multiplications. Each of the various algorithms for the inversion of a matrix requires between n^3 and 8 * n^3 multiplications. In each case, the requirement for a complex matrix is 4 * (real); an isomorph of complex matrix 8 * (real). The saving would be significant; but, not large for today’s computers. The routines that depend upon the cross-copy subroutine require only 4 * (real) for the isomorph of a complex matrix. No potential saving here. Depending upon the desired precision of the resulting eigenvalues and upon the individual symmetric matrix, the Jacobi algorithm requires on the order of 20 n^3 multiplications; the real isomorph of a Hermitain matrix 8 * (real). To maintain swift convergence, when operating upon a Hermitian matrix, the Jacobi algorithm for the eigenpairs (eigenvalue and eigenvector) would have to perform separate kills for the real or imaginary component of a given element. Hence, it would require 8 * (real). No potential saving here. On the other hand, the logistics of implementing a complex library (in addition to the real library) would be considerable; hardly justified because the slowest routines – Jacobi and Gramm-Schmidt – would not benefit.
Since neither of these two theorems has a real-matrix counterpart, we have to place them here, all by their lonesome.
The trace of a general complex matrix is complex. However ---
Lemma: Each of the elements on the principal-diagonal of a Hermitian matrix is purely real.
Proof: Hint: The Hermitian matrix is the complex counterpart of a symmetric matrix.
Theorem: The trace of a Hermitian matrix is purely real.
Proof is obvious.
Lemma: Each of the elements on the principal-diagonal of a skew-Hermitian matrix is purely imaginary.
Proof: Hint: The skew-Hermitian matrix is the complex counterpart of a skew-symmetric matrix.
Theorem: The trace of a skew-Hermitian matrix is purely imaginary.
Proof is obvious.
The proofs of each of the foregoing two lemmata and two theorems could be based upon the observation that the purely-imaginary part of a Hermitian matrix is skew-symmetric while the real-part of a skew-Hermitian matrix is skew-symmetric and upon the corresponding lemma and theorem for the skew-symmetric matrix.
[I added this section in February 2006.]
Contrast this development of the Jordan factorization – which is obtained from the Cholesky factorization – with that of the Dirac factorization – which is obtained from the symmetric or Hermitian factorization.
A polynomial is said to be real; if each of its coefficients is real. However, even a real polynomial may have complex
roots. For example, the real polynomial x^2 + 1 = 0 has the complex roots {i, -i}.
Definition: The characteristic equation of a matrix A is defined as the equation 0 = det(A – Ix).
Definition: The eigen-values of a matrix are the set of the roots of its characteristic equation.
These are the same definitions as we have been using all along; however, now we are generalizing the scope to any general real or general complex matrix
Remarkably, not only does a symmetric matrix (which, by definition, is real) have eigen-values, which are guaranteed to be real; but so does a Hermitian matrix (which, by definition, is complex). We have proven this property of these matrices, already. Here, we present a neat easy alternative proof.
Theorem: Each of the eigen-values of a Hermitian matrix H is real.
Proof: Factor the matrix as H = U D Utr, where U is an unitary matrix of the eigen-vectors and D is a diagonal matrix of the eigen-values. Pre-multiply by Utr and post-multiply by U, to solve for D as D = Utr H U. Transpose, to obtain Dtr = Utr Htr U = Utr H U = D. Hence, D is Hermitian. The diagonal of a Hermitian matrix is, of necessity, real. QED.
Corollary: Each of the eigen-values of a symmetric matrix is real.
Proof: A symmetric matrix is equivalent to a Hermitian matrix, with each of its imaginary elements zero. Thus, the corollary is a special case of the theorem. QED.
However, the eigen-vectors of a general real matrix may be complex, as may be the eigen-values or eigen-vectors of a general complex matrix. Since, these eigen-values or eigen-vectors may be complex, we will consider only complex matrices, from now on. With the understanding, that a real matrix is equivalent to a complex matrix, with each of its imaginary elements zero.
Definition: An operator T is said to be nilpotent of order q, on a given vector space V; provided that q is the least natural number for which T^q v = 0, for each v in V.
In the context of matrices, T would be a square n by n matrix – that is, a matrix of order n – and V would be a set of matrices with n rows and as many columns as desired. Further, ordinarily, V would be the space of all matrices. Then, the condition simplifies to T^q = 0, the null matrix.
For a symmetric (or Hermitian) matrix S, we had defined the set of one or more eigen-vectors corresponding to the eigen-value x, as a set of linearly independent vectors spanning the null-space of the matrix (S – I x). For convenience, we ortho-normalize these vectors. There always are exactly as many such vectors as the degeneracy of the eigen-value – that is, as the nullity of the matrix (S – I x).
For a non-degenerate real (or complex) matrix A, we need only one eigen-vector to correspond to each eigen-value, which is provided by the single-degree nullity of the matrix (A – I x). However, if the matrix A is degenerate; the matrix (A – I x) may not provide sufficient linearly-independent eigen-vectors, as shown by this example: Let A = (½, 1; 0, ½). Its characteristic polynomial is 0 = det(A – I x) = (½– x)^2. Thus, the eigen-values are {½, ½}. The matrix (A – I x) = (A – I * ½) = (0, 1; 0, 0) provides only the single eigen-vector (0, 1). We have to try harder to find the required additional eigen-vectors.
Theorem: For a given degenerate eigen-value, of degeneracy d, of the real (or complex) matrix A; the matrix (A – I x) is idempotent of at most the degree d.
Proof is obvious.
Thus, we employ the null-space of the matrix (A – I x)^d for the additional eigen-vectors. They are known as super eigen-vectors.
Corollary: The nullity of the matrix (A – I x)^d is exactly d.
Proof is obvious.
Definition: The diagonal immediately above the principal diagonal is called the super-diagonal. The diagonal immediately below the principal diagonal is called the sub-diagonal.
Theorem: A matrix N, of order n, each of whose elements is zero, except that each of its super-diagonal elements is one, is nilpotent of order n
Proof is obvious.
Corollary: A nilpotent matrix N, of order n, is nilpotent of order at most n.
Proof is obvious.
Theorem: The transpose of a nilpotent matrix N, of nilpotent order q, is nilpotent of order q.
Proof is obvious.
Definition: A Jordan block is matrix equal to (I x + N), where the matrix N has each of its elements zero, except some of the elements on its super-diagonal one.
Definition: A Jordan matrix J is comprised of Jordan blocks, with distinct x’s, arranged along its principal diagonal.
Lemma: A power of a Jordan block may be computed by means of the binomial theorem.
Proof is obvious.
Theorem: A power of a Jordan matrix is comprised of the corresponding power of each of its Jordan blocks, arranged along its principal diagonal.
Proof is obvious.
Write the Jordan matrix J as J = D + N’, where D is a diagonal matrix and N’ is the remainder matrix.
Theorem: The matrix N’ is nilpotent of an order equal to the maximum of the orders of the N of the Jordan blocks.
Proof is obvious.
Corollary: A power of the Jordan matrix J may be computed by the binomial theorem.
Proof is obvious.
Theorem: Given a non-singular matrix M, for any natural number m, the m-th power of the matrix (M J Minv) is given by (M J Minv)^m = M J^m Minv.
Proof is obvious.
[I was inspired by the Jacobi algorithm, in the design of the following algorithm. Does is work, really? I will not believe it until I write a computer subroutine which implements it. Is it original with me? Probably so; however, I do not know for certain. Look for it yourself and tell me.]
It is our goal to obtain a factorization of any complex matrix A in the form A = M J Minv, where M is a non-singular matrix and J is a Jordan normal form.
In general, the similarity transformation employing the matrix M which consists of a 2-by-2 matrix superimposed upon and identity matrix casts a shadow in a #-pattern. Since here each of the matrices is an upper-triangular matrix, the shadow can be cast only to the right and upwards from the three elements in the 2-by-2 matrix.
In each of the following two lemmata, the shadow is cast up and to the right from the element in the Cholesky matrix, selected for processing. Any additional shadow is described individually.
Consider the 2-by-2 Cholesky matrix Ch = (a, b; 0, d).
Lemma: Non-degenerate Ch -- that is, d is not equal to a. The matrix M = (1, b / (a – d); 0, 1) kills the element b. The inverse of M is Minv = (1, - b / (a – d); 0, 1). The diagonal elements a and d remain invariant. Since there is no additional shadow, this matrix M works for either direction along the diagonal.
Proof: Hint: Multiply-out the product M Ch Minv. It is equal to (a, 0; 0, d). QED.
Lemma: Degenerate Ch – that is, d is equal to a --, with b equal to neither one nor zero. The matrix M = (1, sqrt(1 – b^2); 0, b) normalizes the b to become one. The inverse of M is Minv = (1, - sqrt(1 – b^2) / b, 0, 1 / b). The diagonal elements a and d remain invariant. Since there is an additional shadow cast to the right from the location of the element b in the matrix M, this matrix M works only for the right and down direction along the super-diagonal. The matrix M = (1 / b, sqrt(1 – b^2) / b; 0, 1) casts its additional shadow up from the location of the element 1/b in the matrix M. Hence, this matrix M works only for the left and up direction along the super-diagonal.
Proof: Hint: Multiply-out the product M Ch Minv. It is equal to (a, 1; 0, a). QED.
Pray, observe that for a real b, if it is either greater than one or less than minus one, the aforementioned square-root of (1 – b^2) will be purely imaginary. Of course, if b already is complex; this square-root likewise will be complex. It is this lemma which places the Jordan form of a real matrix into the complex domain. In this hyperlink, we show how to obtain the square-root of a complex number.
Lemma: Degenerate Ch – that is, d is equal to a --, with b equal to either one or zero. There is nothing that can be accomplished. Hence, leave it alone. Since obviously there is no shadow cast by doing nothing, this lemma works for either direction along the super-diagonal.
Now, consider the 3-by-3 degenerate Cholesky matrix Ch = (x, 1, b; 0, x, 1; 0, 0, x).
Lemma: The matrix M = (1, 0, 0; 0, 1, b; 0, 0, 1) kills the element b. The inverse of M is Minv = (1, 0, 0; 0, 1, -b; 0, 0, 1). The diagonal elements x remain invariant. Since the shadow is cast up from the selected element and to the right from the element below the selected element, this matrix M works only for the right and down direction along the diagonal. The matrix M = (1, - b, 0; 0, 1, 0; 0, 0 1) casts its shadow to the right from the selected element and up from the element to the left of the selected element. Hence, this matrix M works only for the left and up direction along the diagonal.
Proof: Hint: Multiply-out the product M Ch Minv. It is equal to (x, 1, 0; 0, x, 1; 0, 0, x). QED.
Finally, consider any order Cholesky matrix Ch.
By means of suitable permutations, arrange it so that each set of degenerate eigen-values comprises a contiguous block. Since complex numbers are not ordered, it is impossible to sort the complex eigen-values. Use your ingenuity as to how to interpret the requirement stated in the pen-ultimate sentence. Alternatively, skip this step -- it is not crucial to the fulfillment of this algorithm. Pray, see the footnote number 01.
Theorem/algorithm: Factorization of the Cholesky into the Jordan, namely Ch = M J Minv. Proceed along each diagonal – either to the right and down xor to the left and up – beginning with the super-diagonal. Then, proceed along successive higher diagonals. For each element, employ the applicable lemma. Since a single pass converts the Cholesky form to the Jordan normal form, there is no question of convergence. Approximately (2/3) n^3 complex multiplications are required for this algorithm.
Proof: It is left as an exercise for the reader to show that the shadow is cast as mentioned in each of the foregoing lemmata.
If we were to employ the definitions of the eigen-value and eigen-vector directly; we would require approximately (1/6) n^4 complex multiplications, to convert the Cholesky to the Jordan factorizations.
Corollary: Factorization of a general complex matrix A into the Jordan, namely A = M J Minv. We provide a constructive proof.
Proof: We already know how to factor a general complex matrix A into the Cholesky, namely A = U Ch Utr, where U is an appropriate unitary matrix. The theorem provides us with the factorization of the Ch into the Jordan, namely Ch = M’ J M’inv. Combine these equations, to yield A = U (M’ J M’inv) Utr = (U M’) J (M’inv Utr) = (U M’) J (M’ U)inv = M J Minv, where the matrix M be given by M = U M’. QED.
Corollary: The matrix M is non-singular.
Proof: The matrix M’, being a product of the non-singular matrices from the lemmata, is non-singular. Any unitary matrix is non-singular. Since M is the product of an unitary matrix U and the matrix M’, each of which is non-singular, so is M non-singular. QED.
Lemma: For a given eigen-value x of the Jordan matrix J, the null-space vector Vo of the matrix (J – I x) is a one in the position of the x, with the remaining elements zero.
Proof is obvious.
Theorem: The eigen-vector of the matrix A, whose Jordan factorization is A = M J Minv, corresponding to the eigen-value x, is V = Minv Vo.
Proof: By the propagation of null-spaces, we have that the vector V is in the null-space of (A – I x). Hence, by the definition of an eigen-vector, V is the corresponding eigen-vector. QED.
Corollary: The set of eigen-vectors of the matrix A is linearly independent.
Proof: Because the matrix M is non-singular. QED.
In Linear Algebra -- as is usual in Mathematics -- one considers the simplest possible manifestation of a given situation. Thus, we study the eigen-values in terms of 0 = det(A - x I) and the eigen-vectors as the null-space of (A - x I). In some applications -- classical mechanics and electricity -- a more complicated version of the eigen-pair problem arises. We have a pair of positive-definite symmetric matrices -- conventionally written as A and B -- in the expression (A - x B). That is. the eigen-values come from 0 = det(A - x B) and the eigen-vectors are the null-space of (A - x B). One may reduce this new problem to the usual formulation by pre-multiplication with the identity in the guise of (B Binv), to yield the form (B Binv) (A - x B) = B ((Binv A) - x I). Pray, remember that just because the matrices A and B each is positive-definite symmetric says nothing regarding the quotient (Binv A). Indeed, this quotient is just a general real matrix. We have to perform the Jordan factorization upon it. Namely, let (Binv A) = M J Minv. Substitute, to obtain the form B ((M J Minv) - x I) = (B M) (J - x I) Minv. Now, we obtain the eigen-values from 0 = det(J - x I). They propagate unchanged to the form 0 = det(A - x B). The eigen-vectors of the form (J - x I) propagate to those of the form (A - x B), by the techniques described in the foregoing theorem. Pray, observe, that when the Cholesky matrix is non-degenerate, the Jordan matrix is a diagonal matrix. However, when the Cholesky matrix is degenerate, the Jordan normal form has some ones in its super-diagonal. As a result we obtain super eigen-vectors. They are not axial-vectors. furthermore, they may be complex. In the application, a degeneracy is called "resonance". Pray, see tensor rank parity for a presentation of such a model.
Theorem: The det(A) = det(J) = product of the eigen-values. The trace(A) = trace(J) = sum of the eigen-values.
Proof: The Jordan factorization of A = M J Minv is a generalized similarity transformation. The determinant of a matrix is invariant under a generalized similarity transformation. The trace of a matrix is invariant under a generalized similarity transformation. QED.
Degrees of freedom.
There is a contradiction between the definition of a super eigen-vector as an axial vector and the convention to have the elements, corresponding to a degeneracy, on the super-diagonal of the Jordan matrix be one. Did you notice? The easiest way to demonstrate this discrepancy is by counting-up the degrees of freedom.
The Cholesky matrix has (n + 1) n / 2 complex elements, for a total of (n + 1) n degrees of freedom. It has to equal the sum of the two following:
An eigen-value, being a complex number, has two degrees of freedom. Since there are n eigen-values, we have 2 n degrees of freedom here.
The M matrix for the factorization of the Cholesky Ch into the Jordan Ch = M J Minv has (n + 1) n / 2 complex elements, for a total of (n + 1) n degrees of freedom. However, since each of these n vectors is axial, we have to subtract one for the magnitude being arbitrary – usually taken as one – and subtract another one for the phase-angle being arbitrary – usually taken as zero. Hence, the net is (n + 1) n – 2 n = (n – 1) n.
If these were all there was to it; we would be in balance.
However, in the degeneracy of the Jordan matrix, it is conventional to fix each element on the super-diagonal to be one. We loose two degrees of freedom, thereby, for each such element. To compensate, we have to allow each super eigen-vector to be a vector – rather than an axial vector.
Theorem: The eigen-values of the Jordan form are those already provided by the Cholesky form.
Proof: Observe that in each case, the diagonal elements – that is, the eigen-values – remain invariant. Thus, the Cholesky factorization already finds the eigen-values of the Jordan normal form, regardless of any possible degeneracies. QED.
Corollary: Each eigen-value of the Jordan factorization of any real matrix is real.
Proof: The Cholesky form of a real matrix is real. QED.
We already know that the Cholesky factorization is unique up to degeneracies and permutations.
Theorem: The Jordan normal form, for any given matrix A, is unique, up to degeneracies and permutations.
Proof: The Jordan factorization inherits its uniqueness from that of the Cholesky factorization..
Corollary: The vectors of M corresponding to each non-degenerate eigen-value – that is, the eigen-vectors – are unique axial vectors. The space spanned by the eigen-vectors corresponding to each degenerate eigen-value is unique.
Proof is obvious.
Theorem: Any non-degenerate complex matrix may be written as a product of two Hermitian matrices.
Proof: Write the Jordan factorization of A as A = M J Minv. Since A is, by hypothesis, non-degenerate, the Jordan matrix J becomes a diagonal matrix D, which may be complex. Find the principal square-root of D and call it D’, which is even more likely to be complex. Then, write A as A = M D Minv = M (D’ D’) Minv = (M D’) (D’ Minv) = (M D’) (Mtr Mtrinv) (D’ Minv) = (M D’ Mtr) (Minvtr D’ Minv) = H1 H2, where H1 = M D’ Mtr and H2 = Minvtr D’ Minv are two Hermitian matrices. QED.
Corollary: Any non-degenerate positive real matrix A may be written as a product of two symmetric matrices.
Proof is the same as of the theorem. Observe that the square-root of a positive number is real.
Tensors are partitioned, by the parity of their tensor-rank, into two classes: even or odd. As we generalize a particular concept, the tensor-rank of the tensor that represents the concept increases, always keeping within its class. Some examples of tensors are: 0) A scalar is of tensor-rank zero. 1) A vector is of tensor-rank one. and 2) A square matrix is of tensor-rank two. Higher tensor-rank tensors are encountered in Riemannian Geometry and General Relativity.
In Linear Algebra, we write a tensor of tensor-rank zero as a scalar times the identity matrix. We write a tensor of tensor-rank one as either an n-by-1 or 1-by-n matrix. Finally, we write a tensor of tensor-rank two as a square matrix.
Stress is defined as the force per unit area. Strain is defined as the relative elongation. Young’s elastic modulus is defined as the quotient of stress divided by strain. Stress and strain each is of even tensor-rank.
Let us begin by considering both the strain and the stress to be a tensor of tensor-rank zero; that is, a scalar. Since division by zero is not permitted, we have to exclude a strain of zero. To keep its domain locally compact, we take the domain as the open interval (0, oo). Since Young’s modulus is approximately a constant, the domain of the stress also is the same interval. In descriptive words, we may say that each of the stress and the strain is strictly-positive. We may write Young’s elastic equation as 0 = stress – Y strain, where Y is the Young’s modulus.
Let us generalize. Stress becomes a matrix, which we write as A. Strain also becomes a matrix, which we write as B. By inheritance, each of these matrices is positive-definite – our terminology for a matrix, each of whose eigen-values is strictly-positive. Not to get ahead of ourselves, we take these matrices as symmetric. As a result, even if either is degenerate, they can support neither oscillation nor resonance. Now, we may write Young’s elastic equation in two parts. For the eigen-values, we write 0 = det(A – x B). The roots x are the eigen-values – they are analogous to Young’s modulus. Then, each corresponding eigen-vector is in the null-space of (A – x B). We solve this last equation by means of the Jordan factorization of the quotient (Binv A), which is a general real matrix. If this quotient matrix is degenerate, we finally obtain both oscillation and resonance.
In the form (A – x B), the A is called the stress and the B is called the strain.
Pray, answer the following questions in the order that they are presented; because, each subsequent question is dependent upon the correct answer of its predecessor.
Part I scalar
What is the parity of the tensor-rank of A and B? Ans. Even.
Why must B be definite? Ans. Division by zero is not allowed.
Why must B be positive? Ans. The domain has to be locally compact.
Could B be negative-definite? Why? Ans. Yes. There is no significant difference between the positive-definite and the negative-definite..
If “No.”; why? Else, why not use negative-definite? Ans. It is easier to use positive-definite rather than negative-definite.
Why must A be positive-definite? Ans. A is proportional to B.
Part II matrices
Could B and A be vectors? Why? Ans. No. The even parity of the tensor-rank always has to be preserved.
Why must B and A be square matrices? Ans. A definite – that is, non-singular – matrix must be square.
Why do we want B and A to be symmetric? What would be the consequence otherwise? Ans. Otherwise either A or B could support oscillation by itself.
What is the ramification of the quotient matrix (Binv A) being a general real matrix, rather than a symmetric matrix? Ans. Oscillation results from any degeneracy of a general real matrix.
Why should the forces be applied along the eigen-vectors of the quotient matrix? Ans. Stability.
Part III Consider the factorization A = O a Otr and B = Q b Qtr, where O and Q are ortho-normal matrices and a and b are diagonal matrices, each of whose elements is strictly positive. By convention, we sort a and b in decreasing order. Pray, see the Field of symmetric matrices which share a common ortho-normal factor for the background.
Why is it desirable for the product matrix P = Qtr O to be a permutation matrix? Ans. Superior stability.
Why is it desirable for P to be that specific permutation matrix I, the identity matrix. Ans. To equalize the strain.
This is not part of the quiz. It is just a rhetorical question. Is there another way, within Linear Algebra, to model the relationship of strain caused by stress? Aye, forsooth there is! We may consider the stress to be the input to a channel and strain to be the output. Then, Young's elastic modulus generalizes to the correlation-coefficient, when both the stress and strain are normalized -- what statisticians call the "z-scale" -- with a mean of zero and a standard-deviation of one. Pray, see the multi-dimensional linear regression discussion.
Footnote number 01.
All right. I see that I will have to explain the nitty-gritty of sorting and comparing numbers.
Snce integers are totally ordered, they may be sorted. And equality of a pair of integers may be tested in an obvious way.
Since real numbers also are totally ordered, they likewise may be sorted. However, because of inevitable round-off errors, a straight-forward
equality test is inappropriate. Instead, one has to test the absolute-value of the difference between a pair of numbers for being less than
or equal to some test criterion. This test criterion has to be large enough to allow for the round-off errors inherent in floating point
calculations. It may have to be even larger, to account for instrumental errors in the measurements.
Since complex numbers are not ordered, they cannot be sorted. We have to find a real number associated with the given complex number.
The obvious candidate is the magnitude of the complex number. Let us major-sort the magnitudes of the complex numbers and compare the
absolute value of the difference between the magnitudes of a pair of complex numbers against a test criterion, just as we did with real numbers.
We are not finished, yet. Just because a pair of complex numbers has their magnitudes substantially equal does not make the complex numbers
equal. We have to minor-sort their phase-angles, then test the absolute value of the difference of their phase-angles against another test
criterion. Do not forget that minus 179 degrees is close to plus 179 degrees. A pair of complex numbers can be considered substantially
equal only when both the absolute value of the difference between their magnitudes is less than or equal to a test criterion and the absolute value
of the difference between their phase angles is less than or equal to a test criterion.
Copyright (c) 2003, 4, 6 by R.I. ‘Scibor-Marchocki. Last revised Friday 03-rd March 2006. Typographical errors corrected and hyper-links added Thursday 23-rd March 2006.