예제: QR 행렬 인수 분해

함수 > 벡터 및 행렬 > 행렬 인수 분해 > 예제: QR 행렬 인수 분해

예제: QR 행렬 인수 분해

QR 함수를 사용하여 QR 행렬 인수 분해를 수행합니다.

• 부울 비교를 수행할 때 논리적 불일치를 피하려면 계산 옵션 드롭다운 목록에서 근사 등호를 활성화합니다.

• 예제에서는 입력으로 복소수 행렬을 사용하지만 이 함수에는 입력으로 실수 행렬도 사용할 수 있습니다.

주축을 사용한 QR 인수 분해

1. 치수 m x n(m > n)인 실수 행렬 M1을 정의합니다.

2. 주축의 활성화/비활성화를 제어하는 인수 p를 설정합니다.

3. QR 함수를 사용하여 행렬 M1의 QR 행렬 인수 분해를 수행합니다.

기본 함수 QR(M1)은 QR(M,1).과 동일합니다.

4. M1 x P1 = Q1 x R1임을 보여줍니다.

<region id="ID0EZAAFB" actualWidth="167.83666666666665" actualHeight="81.4" top="1209.6000000000001" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="16"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M1</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">P1</id> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0EHBAFB" actualWidth="165.74" actualHeight="81.4" top="1209.6000000000001" left="211.20000000000005" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="18"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Q1</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">R1</id> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0E1BAFB" actualWidth="153.23666666666668" actualHeight="25.6" top="1286.4" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="20"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <equal /> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M1</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">P1</id> </apply> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Q1</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">R1</id> </apply> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
관계는 논리적으로 참입니다.

5. submatrix 함수를 사용하여 행렬 M2(m < n)을 추출한 다음 QR 함수를 적용합니다.

6. M2 x P2 = Q2 x R2임을 보여줍니다.

<region id="ID0EHIAFB" actualWidth="69.890000000000015" actualHeight="36.800000000000004" top="1516.8000000000002" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="26"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">P2</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m2</id> <real>0</real> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0EVIAFB" actualWidth="71.38666666666667" actualHeight="36.800000000000004" top="1516.8000000000002" left="134.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="28"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">Q2</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m2</id> <real>1</real> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0EDJAFB" actualWidth="70.816666666666677" actualHeight="36.800000000000004" top="1516.8000000000002" left="230.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="30"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">R2</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m2</id> <real>2</real> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EWJAFB" actualWidth="153.23666666666668" actualHeight="25.6" top="1555.2000000000003" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="32"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <equal /> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M2</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">P2</id> </apply> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Q2</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">R2</id> </apply> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
관계는 논리적으로 참입니다.

7. submatrix 함수를 사용하여 행렬 M3(m = n)을 추출한 다음 QR 함수를 적용합니다.

8. M3 x P3 = Q3 x R3임을 보여줍니다.

<region id="ID0E2PAFB" actualWidth="69.890000000000015" actualHeight="36.800000000000004" top="1785.6000000000004" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="38"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">P3</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m3</id> <real>0</real> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0EJQAFB" actualWidth="71.38666666666667" actualHeight="36.800000000000004" top="1785.6000000000004" left="134.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="40"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">Q3</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m3</id> <real>1</real> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0EXQAFB" actualWidth="70.816666666666677" actualHeight="36.800000000000004" top="1785.6000000000004" left="230.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="42"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">R3</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m3</id> <real>2</real> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EKRAFB" actualWidth="153.23666666666668" actualHeight="25.6" top="1824.0000000000002" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="44"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <equal /> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M3</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">P3</id> </apply> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Q3</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">R3</id> </apply> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
관계는 논리적으로 참입니다.

주축을 사용하지 않는 QR 인수 분해

1. 주축을 비활성화한 다음 행렬 M1 (m > n)에 QR 함수를 적용합니다.

2. M1 = Q10 x R10임을 보여줍니다.

<region id="ID0E4XAFB" actualWidth="140.69333333333333" actualHeight="81.4" top="2380.8" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="52"> <eval xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M1</id> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0ELYAFB" actualWidth="170.95" actualHeight="81.4" top="2380.8" left="211.20000000000005" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="54"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Q10</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">R10</id> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0E5YAFB" actualWidth="142.05333333333334" actualHeight="25.6" top="2457.6000000000004" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="56"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <equal /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M1</id> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Q10</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">R10</id> </apply> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
관계는 논리적으로 참입니다.

3. 주축을 비활성화한 다음 행렬 M2 (m < n)에 QR 함수를 적용합니다.

4. M2 = Q20 x R20임을 보여줍니다.

<region id="ID0EC5AFB" actualWidth="87.346666666666664" actualHeight="36.800000000000004" top="2649.6000000000004" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="60"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">Q20</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m20</id> <real>0</real> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0EQ5AFB" actualWidth="86.776666666666671" actualHeight="36.800000000000004" top="2649.6000000000004" left="134.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="62"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">R20</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m20</id> <real>1</real> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0ED6AFB" actualWidth="142.05333333333334" actualHeight="25.6" top="2688.0000000000005" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="64"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <equal /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M2</id> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Q20</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">R20</id> </apply> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
관계는 논리적으로 참입니다.

5. 주축을 비활성화한 다음 행렬 M3 (m = n)에 QR 함수를 적용합니다.

6. M3 = Q30 x R30임을 보여줍니다.

<region id="ID0EGEBFB" actualWidth="87.346666666666664" actualHeight="36.800000000000004" top="2880.0000000000005" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="68"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">Q30</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m30</id> <real>0</real> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0EUEBFB" actualWidth="86.776666666666671" actualHeight="36.800000000000004" top="2880.0000000000005" left="134.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="70"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">R30</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m30</id> <real>1</real> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EHFBFB" actualWidth="142.05333333333334" actualHeight="25.6" top="2918.4000000000005" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="72"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <equal /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M3</id> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Q30</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">R30</id> </apply> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
관계는 논리적으로 참입니다.