By Sudipto Banerjee

"Linear algebra and the examine of matrix algorithms became primary to the advance of statistical types. utilizing a vector-space method, this e-book presents an figuring out of the key options that underlie linear algebra and matrix research. each one bankruptcy introduces a key subject, reminiscent of infinite-dimensional areas, and gives illustrative examples. The authors research fresh advancements in diverseRead more...

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**Example text**

Only nonzero numbers are allowed to be pivots. If a coefficient in a pivot position is 0, then that row is interchanged with some row below the pivotal 36 SYSTEMS OF LINEAR EQUATIONS below it to produce a nonzero pivot. ) Unless it is 0, we take the first element of the first row in the augmented matrix as the first pivot. Otherwise, if it is 0, we can always rearrange the system of equations so that an equation with a nonzero coefficient of x1 appears as the first equation. This will ensure that the first element in the first row of the augmented matrix is nonzero.

Ajn1 ) is the j-th column of A21 and (ajn1 +1 , . . , ajn ) is the j-th column of A22 . Therefore, A= A11 A21 A12 A22 implies that A = A11 A12 A21 A22 . 12) The matrix A is now n × m with A11 being n1 × m1 , A12 being n2 × m1 , A21 being n1 × m2 and A22 being n2 × m2 . Note that A is formed by transposing each of the four block matrices and, in addition, switching the positions of A12 and A21 . We say that two partitioned matrices A and B are conformably partitioned for addition when their corresponding blocks have the same order.

Proof. Since Q is a permutation matrix, we can write Q = [e∗j1 : · · · : e∗jn ]. Postmultiplication by Q permutes the columns of P as P Q = [P ∗j1 : · · · : P ∗jn ]. As the columns of P are a permutation of the identity matrix, the columns of P Q are also a permutation of the identity matrix. Therefore, P Q is a permutation matrix. 22 MATRICES, VECTORS AND THEIR OPERATIONS If we have permuted the rows (columns) of A by pre- (post-) multiplication with a permutation matrix P , then how can we “undo” the operation and recover the original A matrix?