Arrowhead like matrix - Alchetron, The Free Social Encyclopedia

In mathematics, an arrowhead-like matrix is a matrix of the form

[ X m × m B o , m × n 0 m × k C o , n × m D n × n C v , n × k 0 k × m B v , k × n 0 k × k ]

where X is a diagonal matrix.

Inverses

When n ≥ k, the arrowhead-like matrix has its inverse matrix

[ X m × m B o , m × n 0 m × k C o , n × m D n × n C v , n × k 0 k × m B v , k × n 0 k × k ] − 1 = [ X − 1 + X − 1 B o ( D − D C v [ B v D C v ] − 1 B v D ) C o X − 1 − X − 1 B o ( D − D C v [ B v D C v ] − 1 B v D ) − X − 1 B o D C v [ B v D C v ] − 1 − ( D − D C v [ B v D C v ] − 1 B v D ) C o X − 1 D − D C v [ B v D C v ] − 1 B v D D C v [ B v D C v ] − 1 − [ B v D C v ] − 1 B v D C o X − 1 [ B v D C v ] − 1 B v D − [ B v D C v ] − 1 ] D = ( D − C o X − 1 B o ) − 1

special cases:

k = 0:

[ X B o C o D ] − 1 = [ X − 1 + X − 1 B o D C o X − 1 − X − 1 B o D − D C o X − 1 D ]

k = n:

[ X B o 0 C o D C v 0 B v 0 ] − 1 = [ X − 1 0 − X − 1 B o B v − 1 0 0 B v − 1 − C v − 1 C o X − 1 C v − 1 − C v − 1 ( D − C o X − 1 B o ) B v − 1 ]

References

Arrowhead-like matrix Wikipedia

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