powers of a square matrix!
and matrix form of a linear systemEPC4 3.4.43LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkobWFjdGlvbkdGJDYkLUkobWZlbmNlZEdGJDYoLUYjNictRjY2LVEhRidGOUY7Rj5GQEZCRkRGRkZIRkpGTS1JJ210YWJsZUdGJDY4LUkkbXRyR0YkNiktSSRtdGRHRiQ2KS1GIzYlLUkjbW5HRiQ2JFEiMkYnRjkvJStleGVjdXRhYmxlR0Y9RjlGY28vJSlyb3dhbGlnbkdGWS8lLGNvbHVtbmFsaWduR0ZZLyUrZ3JvdXBhbGlnbkdGWS8lKHJvd3NwYW5HUSIxRicvJStjb2x1bW5zcGFuR0ZdcC1GW282KS1GIzYmLUY2Ni1RKiZ1bWludXMwO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZocC1GYG82JEZdcEY5RmNvRjlGY29GZW9GZ29GaW9GW3BGXnBGYHBGY29GZW9GZ29GaW8tRmhuNilGYHBGam5GYHBGY29GZW9GZ29GaW8tRmhuNilGYHBGYHBGam5GY29GZW9GZ29GaW9GY28vJSZhbGlnbkdRJWF4aXNGJy9GZm9RKWJhc2VsaW5lRicvRmhvUSdjZW50ZXJGJy9Gam9RJ3xmcmxlZnR8aHJGJy8lL2FsaWdubWVudHNjb3BlR0YxLyUsY29sdW1ud2lkdGhHUSVhdXRvRicvJSZ3aWR0aEdGXXIvJStyb3dzcGFjaW5nR1EmMS4wZXhGJy8lLmNvbHVtbnNwYWNpbmdHUSYwLjhlbUYnLyUpcm93bGluZXNHUSVub25lRicvJSxjb2x1bW5saW5lc0dGaHIvJSZmcmFtZUdGaHIvJS1mcmFtZXNwYWNpbmdHUSwwLjRlbX4wLjVleEYnLyUqZXF1YWxyb3dzR0Y9LyUtZXF1YWxjb2x1bW5zR0Y9LyUtZGlzcGxheXN0eWxlR0Y9LyUlc2lkZUdRJnJpZ2h0RicvJTBtaW5sYWJlbHNwYWNpbmdHRmVyRldGY29GOUY5L0krbXNlbWFudGljc0dGJFEnTWF0cml4RicvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGW3QvJSthY3Rpb250eXBlR1EucnRhYmxlYWRkcmVzc0YnRldGY29GOQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnRj4vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRkBGPg==If you do this by hand and compare, you will find that 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW5HRiQ2JFEiM0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkctSSNtaUdGJDYlUSJBRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdGOEYvIt is easier to just check that the difference matrix is zero.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JSFHNotice that this tells us how to evaluate all the integer powers of this matrix!
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And formally the exponential would be (we can only use this relation after the first two terms, using the fact that 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)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 means we can "sum" the infinite series easily and express it in terms of a combination of the identity matrix and the matrix A.
Chapter 8 in our textbook is about the matrix exponential, but although it plays a fundamental rolein nonlinear matrix algebra, we have no time for it on this pass through the material.inverse?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This is a "singular matrix" (does not have an inverse) which is signaled by the zero value of the determinant (section 3.6)3.4.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This linear system is already fully reduced. Only the backsubstitution step is required.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