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Matrix : Convergence, Pseudoinverse and Single Value Decomposition
norm:
Remember that the maximal absolute value of any of the terms in D is less than 1, due to the condition .
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Idempotent and nilpotent matrix proofs
So has eigenvalues 1 and 3. We can find a non-singular matrix , such that , where . Then . So we have
Therefore,
.
Therefore,
Problem #6
We know, the singular values of are square roots of eigenvalues of .
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vector spaces
The zero matrix is symmetric. This IS a vector space.
(b) The sum of two non-singular matrices need not be non-singular. Consider, for example, the identity matrix I and the matrix (-I).
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The Jacobian Matrix in the Implicit Function Theorem
, because F_{U}({X},{U}) needs to be non-singular. For a matrix to be
invertible (or non-singular), it has to be a square matrix to start with, that is to say, it has to have the same number of rows and columns.
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Characteristic root of a matrix - Proof
4155 Roots of non singular matrix. Proof Show that if lamda is a characteristic root of a non-singular matrix A, then
lamda^-1 is a characteristic root of A^-1.
Please see the attached file. Detailed solution as a pdf attachemant.
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Module homomorphism proof
This implies that is a non-singular matrix and thus its inverse exists and is also non-singular.
So we have is a non-singular matrix. Therefore, has unique solution . Hence is one-to-one or injective.
Now we show the converse is false.
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Determinants and Adjugate : Proof
We know that:
A^(-1)=adj(A)/det(A)
So:
adj(A)=A^(-1)*det(A)
Now we take the determinant of both sides:
det(adj(A))=det(A^(-1)*det(A))
A is n*n and det(A) is a non-zero number and we know that det(k*A)=k^n*det(A) if A is n*n matrix.
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Properties of condition numbers : Orthogonal Matrices and Eigenvalues
The definition of the condition number of is . So we always consider that is non-singular.
Proof:
1. Since , then by definition, we have
Thus we get .
2. We know, . I claim that .
For any non-zero , we consider .
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Matrices
Verify the special theorem for the Hermitian matrix see attached This provides examples of working with matrices, including eigenvalues and singular values, singular decomposition, and Hermitian matrix.