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Indeed, such a matrix can be reduced, by appropriately adding multiples of the columns with fewer nonzero entries to those with more entries, to a diagonal matrix (without changing the determinant). For such a matrix, using the linearity in each column reduces to the identity matrix, in which case the stated formula holds by the very first ...
The identity matrix is the only idempotent matrix with non-zero determinant. That is, it is the only matrix such that: When multiplied by itself, the result is itself; All of its rows and columns are linearly independent. The principal square root of an identity matrix is itself, and this is its only positive-definite square root. However ...
The determinant of the left hand side is the product of the determinants of the three matrices. Since the first and third matrix are triangular matrices with unit diagonal, their determinants are just 1. The determinant of the middle matrix is our desired value. The determinant of the right hand side is simply (1 + v T u). So we have the result:
In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. [1]If A is a differentiable map from the real numbers to n × n matrices, then
In linear algebra, an invertible complex square matrix U is unitary if its matrix inverse U −1 equals its conjugate transpose U *, that is, if = =, where I is the identity matrix.. In physics, especially in quantum mechanics, the conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted by a dagger ( † ), so the equation above is written
A square matrix with zero diagonal and +1 and −1 off the diagonal, such that C T C is a multiple of the identity matrix. Complex Hadamard matrix: A matrix with all rows and columns mutually orthogonal, whose entries are unimodular. Compound matrix: A matrix whose entries are generated by the determinants of all minors of a matrix. Copositive ...
Since the eight matrices and the identity are a complete trace-orthogonal set spanning all 3×3 matrices, it is straightforward to find two Fierz completeness relations, (Li & Cheng, 4.134), analogous to that satisfied by the Pauli matrices. Namely, using the dot to sum over the eight matrices and using Greek indices for their row/column ...
In fact, a matrix A over a field containing all of the eigenvalues of A (for example, any matrix over an algebraically closed field) is similar to a triangular matrix. This can be proven by using induction on the fact that A has an eigenvector, by taking the quotient space by the eigenvector and inducting to show that A stabilizes a flag, and ...